{"id":1559,"date":"2017-10-27T16:32:25","date_gmt":"2017-10-27T16:32:25","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/relativistic-energy\/"},"modified":"2017-11-08T03:27:30","modified_gmt":"2017-11-08T03:27:30","slug":"relativistic-energy","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/relativistic-energy\/","title":{"raw":"Relativistic Energy","rendered":"Relativistic Energy"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Compute total energy of a relativistic object.<\/li>\n<li>Compute the kinetic energy of a relativistic object.<\/li>\n<li>Describe rest energy, and explain how it can be converted to other forms.<\/li>\n<li>Explain why massive particles cannot reach C.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id882829\">\n<div class=\"bc-figcaption figcaption\">The National Spherical Torus Experiment (NSTX) has a fusion reactor in which hydrogen isotopes undergo fusion to produce helium. In this process, a relatively small mass of fuel is converted into a large amount of energy. (credit: Princeton Plasma Physics Laboratory)<\/div>\n<p><span data-type=\"media\" data-alt=\"This photo shows the outside of the fusion reactor of the National Spherical Torus Experiment at the Princeton Plasma Physics Laboratory. The reactor, which sits in a large room, is connected to numerous tubes and instruments.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_29_06_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"This photo shows the outside of the fusion reactor of the National Spherical Torus Experiment at the Princeton Plasma Physics Laboratory. The reactor, which sits in a large room, is connected to numerous tubes and instruments.\" width=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1985448\">A tokamak is a form of experimental fusion reactor, which can change mass to energy. Accomplishing this requires an understanding of relativistic energy. Nuclear reactors are proof of the conservation of relativistic energy.<\/p>\n<p id=\"import-auto-id1284621\">Conservation of energy is one of the most important laws in physics. Not only does energy have many important forms, but each form can be converted to any other. We know that classically the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is altered to include the possibility of mass changing to energy, as in the reactions that occur within a nuclear reactor. Relativistic energy is intentionally defined so that it will be conserved in all inertial frames, just as is the case for relativistic momentum. As a consequence, we learn that several fundamental quantities are related in ways not known in classical physics. All of these relationships are verified by experiment and have fundamental consequences. The altered definition of energy contains some of the most fundamental and spectacular new insights into nature found in recent history.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2633718\">\n<h1 data-type=\"title\">Total Energy and Rest Energy<\/h1>\n<p id=\"import-auto-id3137655\">The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define energy to include a relativistic factor.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2636850\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Total Energy<\/div>\n<p id=\"import-auto-id1309522\"><span data-type=\"term\" id=\"import-auto-id2601670\">Total energy <\/span>[latex]E[\/latex] is defined to be<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]E={\\mathrm{\\gamma mc}}^{2},[\/latex]<\/div>\n<p id=\"import-auto-id3163669\">where [latex]m[\/latex] is mass, [latex]c[\/latex] is the speed of light, [latex]\\gamma =\\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}[\/latex], and [latex]v[\/latex] is the velocity of the mass relative to an observer. There are many aspects of the total energy [latex]E[\/latex] that we will discuss\u2014among them are how kinetic and potential energies are included in [latex]E[\/latex], and how [latex]E[\/latex] is related to relativistic momentum. But first, note that at rest, total energy is not zero. Rather, when [latex]v=0[\/latex], we have [latex]\\gamma =1[\/latex], and an object has rest energy.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3099697\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Rest Energy<\/div>\n<p id=\"import-auto-id2644699\"><span data-type=\"term\" id=\"import-auto-id2678807\">Rest energy<\/span> is\n    <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-594\">[latex]{E}_{0}={\\mathrm{mc}}^{2}.[\/latex]<\/div>\n<\/div>\n<p id=\"import-auto-id2994648\">This is the correct form of Einstein\u2019s most famous equation, which for the first time showed that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. The implications of these first two equations regarding relativistic energy are so broad that they were not completely recognized for some years after Einstein published them in 1907, nor was the experimental proof that they are correct widely recognized at first. Einstein, it should be noted, did understand and describe the meanings and implications of his theory.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1516569\">\n<div data-type=\"title\" class=\"title\">Calculating Rest Energy: Rest Energy is Very Large<\/div>\n<p id=\"import-auto-id1599949\">Calculate the rest energy of a 1.00-g mass.<\/p>\n<p id=\"import-auto-id2010072\"><strong>Strategy<\/strong><\/p>\n<p>One gram is a small mass\u2014less than half the mass of a penny. We can multiply this mass, in SI units, by the speed of light squared to find the equivalent rest energy.<\/p>\n<p id=\"import-auto-id1637047\"><strong>Solution<\/strong><\/p>\n<ol id=\"fs-id3459859\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2720059\">Identify the knowns. [latex]m=1\\text{.00}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]; [latex]c=3\\text{.}\\text{00}\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex]<\/li>\n<li id=\"import-auto-id1279713\">Identify the unknown. [latex]{E}_{0}[\/latex]<\/li>\n<li id=\"import-auto-id2632083\">Choose the appropriate equation. [latex]{E}_{0}={\\mathrm{mc}}^{2}[\/latex]<\/li>\n<li>Plug the knowns into the equation.\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}{E}_{0}&amp; =&amp; {\\mathrm{mc}}^{2}=\\left(1.00\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\right)\\left(3.00\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}{\\right)}^{2}\\\\ &amp; =&amp; 9.00\u00d7{\\text{10}}^{\\text{13}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\cdot {\\text{m}}^{2}{\\text{\/s}}^{2}\\end{array}[\/latex]<\/div>\n<\/li>\n<li>Convert units.\n<p id=\"import-auto-id1583964\">Noting that [latex]1\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\cdot {\\text{m}}^{2}{\\text{\/s}}^{2}=1 J[\/latex], we see the rest mass energy is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{E}_{0}=9\\text{.}\\text{00}\u00d7{\\text{10}}^{\\text{13}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id2555003\"><strong>Discussion<\/strong><\/p>\n<p>This is an enormous amount of energy for a 1.00-g mass. We do not notice this energy, because it is generally not available. Rest energy is large because the speed of light [latex]c[\/latex] is a large number and [latex]{c}^{2}[\/latex] is a very large number, so that [latex]{\\mathrm{mc}}^{2}[\/latex] is huge for any macroscopic mass. The [latex]9\\text{.}\\text{00}\u00d7{\\text{10}}^{\\text{13}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex] rest mass energy for 1.00 g is about twice the energy released by the Hiroshima atomic bomb and about 10,000 times the kinetic energy of a large aircraft carrier. If a way can be found to convert rest mass energy into some other form (and all forms of energy can be converted into one another), then huge amounts of energy can be obtained from the destruction of mass.<\/p>\n<\/div>\n<p>Today, the practical applications of <em data-effect=\"italics\"><em data-effect=\"italics\">the conversion of mass into another form of energy<\/em><\/em>, such as in nuclear weapons and nuclear power plants, are well known. But examples also existed when Einstein first proposed the correct form of relativistic energy, and he did describe some of them. Nuclear radiation had been discovered in the previous decade, and it had been a mystery as to where its energy originated. The explanation was that, in certain nuclear processes, a small amount of mass is destroyed and energy is released and carried by nuclear radiation. But the amount of mass destroyed is so small that it is difficult to detect that any is missing. Although Einstein proposed this as the source of energy in the radioactive salts then being studied, it was many years before there was broad recognition that mass could be and, in fact, commonly is converted to energy. (See <a href=\"#import-auto-id2636680\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2636680\">\n<div class=\"bc-figcaption figcaption\">The Sun (a) and the Susquehanna Steam Electric Station (b) both convert mass into energy\u2014the Sun via nuclear fusion, the electric station via nuclear fission. (credits: (a) NASA\/Goddard Space Flight Center, Scientific Visualization Studio; (b) U.S. government) <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2005219\" data-alt=\"Part a of the figure shows a solar storm on the Sun. Part b of the figure shows the Susquehanna Steam Electric Station, which produces electricity by nuclear fission.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_29_06_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows a solar storm on the Sun. Part b of the figure shows the Susquehanna Steam Electric Station, which produces electricity by nuclear fission.\" width=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2836811\">Because of the relationship of rest energy to mass, we now consider mass to be a form of energy rather than something separate. There had not even been a hint of this prior to Einstein\u2019s work. Such conversion is now known to be the source of the Sun\u2019s energy, the energy of nuclear decay, and even the source of energy keeping Earth\u2019s interior hot.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2585048\">\n<h1 data-type=\"title\">Stored Energy and Potential Energy<\/h1>\n<p id=\"import-auto-id1579956\">What happens to energy stored in an object at rest, such as the energy put into a battery by charging it, or the energy stored in a toy gun\u2019s compressed spring? The energy input becomes part of the total energy of the object and, thus, increases its rest mass. All stored and potential energy becomes mass in a system. Why is it we don\u2019t ordinarily notice this? In fact, conservation of mass (meaning total mass is constant) was one of the great laws verified by 19th-century science. Why was it not noticed to be incorrect? The following example helps answer these questions.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1493235\">\n<div data-type=\"title\" class=\"title\">Calculating Rest Mass: A Small Mass Increase due to Energy Input<\/div>\n<p id=\"import-auto-id2784873\">A car battery is rated to be able to move 600 ampere-hours [latex]\\left(A\u00b7h\\right)[\/latex] of charge at 12.0 V. (a) Calculate the increase in rest mass of such a battery when it is taken from being fully depleted to being fully charged. (b) What percent increase is this, given the battery\u2019s mass is 20.0 kg?<\/p>\n<p id=\"import-auto-id3169070\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2721940\">In part (a), we first must find the energy stored in the battery, which equals what the battery can supply in the form of electrical potential energy. Since [latex]{\\text{PE}}_{\\text{elec}}=\\text{qV}[\/latex], we have to calculate the charge [latex]q[\/latex] in [latex]600\\phantom{\\rule{0.25em}{0ex}}A\u00b7h[\/latex], which is the product of the current [latex]I[\/latex] and the time [latex]t[\/latex]. We then multiply the result by 12.0 V. We can then calculate the battery\u2019s increase in mass using [latex]\\Delta E={\\text{PE}}_{\\text{elec}}=\\left(\\Delta m\\right){c}^{2}[\/latex]. Part (b) is a simple ratio converted to a percentage.<\/p>\n<p id=\"import-auto-id2976450\"><strong>Solution for (a) <\/strong><\/p>\n<ol id=\"fs-id2749129\" data-number-style=\"arabic\">\n<li id=\"import-auto-id1610514\">Identify the knowns. [latex]I\\cdot t=\\text{600 A}\\cdot \\text{h}[\/latex]; [latex]V=\\text{12}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex]; [latex]c=3\\text{.}\\text{00}\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex]<\/li>\n<li id=\"import-auto-id2955803\">Identify the unknown. [latex]\\Delta m[\/latex]<\/li>\n<li id=\"import-auto-id2731830\">Choose the appropriate equation. [latex]{\\text{PE}}_{\\text{elec}}=\\left(\\Delta m\\right){c}^{2}[\/latex]<\/li>\n<li id=\"import-auto-id2561455\">Rearrange the equation to solve for the unknown. [latex]\\Delta m=\\frac{{\\text{PE}}_{\\text{elec}}}{{c}^{2}}[\/latex]<\/li>\n<li id=\"import-auto-id1310375\">Plug the knowns into the equation.\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}\\Delta m&amp; =&amp; \\frac{{\\text{PE}}_{\\text{elec}}}{{c}^{2}}\\\\ &amp; =&amp; \\frac{\\text{qV}}{{c}^{2}}\\\\ &amp; =&amp; \\frac{\\left(It\\right)V}{{c}^{2}}\\\\ &amp; =&amp; \\frac{\\left(\\text{600 A}\\cdot \\text{h}\\right)\\left(\\text{12.0 V}\\right)}{\\left(3.00\u00d7{\\text{10}}^{8}{\\right)}^{2}}\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1327415\">Write amperes A as coulombs per second (C\/s), and convert hours to seconds.<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}\\Delta m&amp; =&amp; \\frac{\\left(\\text{600 C\/s}\\cdot \\text{h}\\left(\\frac{\\text{3600 s}}{\\text{1 h}}\\right)\\left(\\text{12.0 J\/C}\\right)}{\\left(3.00\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}{\\right)}^{2}}\\\\ &amp; =&amp; \\frac{\\left(2.16\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{C}\\right)\\left(\\text{12.0 J\/C}\\right)}{\\left(3.00\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}{\\right)}^{2}}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1048320\">Using the conversion [latex]1\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\cdot {\\text{m}}^{2}{\\text{\/s}}^{2}=1\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex], we can write the mass as<\/p>\n[latex]\\Delta m=2.88\u00d7{\\text{10}}^{-\\text{10}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\text{.}[\/latex]\n<\/li>\n<\/ol>\n<p id=\"import-auto-id923710\"><strong>Solution for (b) <\/strong><\/p>\n<ol id=\"fs-id1290816\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2990812\">Identify the knowns. [latex]\\Delta m=2.88\u00d7{\\text{10}}^{-\\text{10}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]; [latex]m=\\text{20.0 kg}[\/latex]<\/li>\n<li id=\"import-auto-id2703315\">Identify the unknown. % change<\/li>\n<li id=\"import-auto-id2604060\">Choose the appropriate equation. [latex]\\text{}\\text{% increase}=\\frac{\\Delta m}{m}\u00d7\\text{100%}\\text{}[\/latex]<\/li>\n<li id=\"import-auto-id2637141\">Plug the knowns into the equation.\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}\\text{% increase}&amp; =&amp; \\frac{\\Delta m}{m}\u00d7\\text{100%}\\\\ &amp; =&amp; \\frac{2.88\u00d7{\\text{10}}^{-\\text{10}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}}{\\text{20.0 kg}}\u00d7\\text{100%}\\\\ &amp; =&amp; 1.44\u00d7{\\text{10}}^{-9}\\text{%}\\text{.}\\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2719499\">Both the actual increase in mass and the percent increase are very small, since energy is divided by [latex]{c}^{2}[\/latex], a very large number. We would have to be able to measure the mass of the battery to a precision of a billionth of a percent, or 1 part in [latex]{10}^{11}[\/latex], to notice this increase. It is no wonder that the mass variation is not readily observed. In fact, this change in mass is so small that we may question how you could verify it is real. The answer is found in nuclear processes in which the percentage of mass destroyed is large enough to be measured. The mass of the fuel of a nuclear reactor, for example, is measurably smaller when its energy has been used. In that case, stored energy has been released (converted mostly to heat and electricity) and the rest mass has decreased. This is also the case when you use the energy stored in a battery, except that the stored energy is much greater in nuclear processes, making the change in mass measurable in practice as well as in theory.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id817296\">\n<h1 data-type=\"title\">Kinetic Energy and the Ultimate Speed Limit<\/h1>\n<p id=\"import-auto-id704456\">Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression [latex]\\frac{1}{2}{\\mathrm{mv}}^{2}[\/latex]. The relativistic expression for kinetic energy is obtained from the work-energy theorem. This theorem states that the net work on a system goes into kinetic energy. If our system starts from rest, then the work-energy theorem is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{W}_{\\text{net}}=\\text{KE}.[\/latex]<\/div>\n<p id=\"import-auto-id1667111\">Relativistically, at rest we have rest energy [latex]{E}_{0}={\\mathrm{mc}}^{2}[\/latex]. The work increases this to the total energy [latex]E={\\mathrm{\\gamma mc}}^{2}[\/latex]. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-888\">[latex]{W}_{\\text{net}}=E-{E}_{0}=\\gamma {\\mathrm{mc}}^{2}-{\\mathrm{mc}}^{2}=\\left(\\gamma -1\\right){\\mathrm{mc}}^{2}.[\/latex]<\/div>\n<p id=\"import-auto-id2676240\">Relativistically, we have [latex]{W}_{\\text{net}}={\\text{KE}}_{\\text{rel}}[\/latex].<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1611892\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Relativistic Kinetic Energy<\/div>\n<p><span data-type=\"term\" id=\"import-auto-id2638011\">Relativistic kinetic energy<\/span> is\n    <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\text{KE}}_{\\text{rel}}=\\left(\\gamma -1\\right){\\mathrm{mc}}^{2}.[\/latex]<\/div>\n<\/div>\n<p id=\"import-auto-id3156287\">When motionless, we have [latex]v=0[\/latex] and<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\gamma =\\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}=1,[\/latex]<\/div>\n<p id=\"import-auto-id2501781\">so that [latex]{\\text{KE}}_{\\text{rel}}=0[\/latex] at rest, as expected. But the expression for relativistic kinetic energy (such as total energy and rest energy) does not look much like the classical [latex]\\frac{1}{2}{\\mathrm{mv}}^{2}[\/latex]. To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for [latex]\\gamma [\/latex] at low velocities gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\gamma =1+\\frac{1}{2}\\frac{{v}^{2}}{{c}^{2}}.[\/latex]<\/div>\n<p id=\"import-auto-id3067754\">A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In some cases, as in the limit of small velocity here, most terms are very small. Thus the expression derived for [latex]\\gamma [\/latex] here is not exact, but it is a very accurate approximation. Thus, at low velocities,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\gamma -1=\\frac{1}{2}\\frac{{v}^{2}}{{c}^{2}}.[\/latex]<\/div>\n<p>Entering this into the expression for relativistic kinetic energy gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-962\">[latex]{\\text{KE}}_{\\text{rel}}=\\left[\\frac{1}{2}\\frac{{v}^{2}}{{c}^{2}}\\right]{\\mathrm{mc}}^{2}=\\frac{1}{2}{\\mathrm{mv}}^{2}={\\text{KE}}_{\\text{class}}.[\/latex]<\/div>\n<p id=\"import-auto-id1362957\">So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when [latex]v\\text{&lt;&lt;}c[\/latex].<\/p>\n<p id=\"import-auto-id3354394\">It is even more interesting to investigate what happens to kinetic energy when the velocity of an object approaches the speed of light. We know that [latex]\\gamma [\/latex] becomes infinite as [latex]v[\/latex] approaches [latex]c[\/latex], so that KE<sub>rel<\/sub> also becomes infinite as the velocity approaches the speed of light. (See <a href=\"#import-auto-id1745737\" class=\"autogenerated-content\">(Figure)<\/a>.) An infinite amount of work (and, hence, an infinite amount of energy input) is required to accelerate a mass to the speed of light.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2836695\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">The Speed of Light<\/div>\n<p id=\"import-auto-id1562844\"><strong>No object with mass can attain the speed of light.<\/strong><\/p>\n<\/div>\n<p id=\"import-auto-id2687489\">So the speed of light is the ultimate speed limit for any particle having mass. All of this is consistent with the fact that velocities less than [latex]c[\/latex] always add to less than [latex]c[\/latex]. Both the relativistic form for kinetic energy and the ultimate speed limit being [latex]c[\/latex] have been confirmed in detail in numerous experiments. No matter how much energy is put into accelerating a mass, its velocity can only approach\u2014not reach\u2014the speed of light.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1745737\">\n<div class=\"bc-figcaption figcaption\">This graph of [latex]{\\text{KE}}_{\\text{rel}}[\/latex] versus velocity shows how kinetic energy approaches infinity as velocity approaches the speed of light. It is thus not possible for an object having mass to reach the speed of light. Also shown is [latex]{\\text{KE}}_{\\text{class}}[\/latex], the classical kinetic energy, which is similar to relativistic kinetic energy at low velocities. Note that much more energy is required to reach high velocities than predicted classically.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3136349\" data-alt=\"In this figure a graph is shown on a coordinate system of axes. The x-axis is labeled as speed v (m\/s). On the x-axis, velocity of the object is shown in terms of the speed of light starting from zero at origin to c, where c is the speed of light. The y-axis is labeled as Kinetic Energy K E (J). On the y-axis, relativistic kinetic energy is shown starting from 0 at origin to 1.0. The graph K sub r e l of relativistic kinetic energy is concave up and moving upward along the vertical line at x equals c. This graph shows that relativistic kinetic energy approaches infinity as the velocity of an object approaches the speed of light. Also shown is that when the speed of the object is equal to the speed of light c the kinetic energy is known as classical kinetic energy, which is denoted as K E sub class.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_29_06_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a graph is shown on a coordinate system of axes. The x-axis is labeled as speed v (m\/s). On the x-axis, velocity of the object is shown in terms of the speed of light starting from zero at origin to c, where c is the speed of light. The y-axis is labeled as Kinetic Energy K E (J). On the y-axis, relativistic kinetic energy is shown starting from 0 at origin to 1.0. The graph K sub r e l of relativistic kinetic energy is concave up and moving upward along the vertical line at x equals c. This graph shows that relativistic kinetic energy approaches infinity as the velocity of an object approaches the speed of light. Also shown is that when the speed of the object is equal to the speed of light c the kinetic energy is known as classical kinetic energy, which is denoted as K E sub class.\" width=\"250\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1553241\">\n<div data-type=\"title\" class=\"title\">Comparing Kinetic Energy: Relativistic Energy Versus Classical Kinetic Energy<\/div>\n<p id=\"import-auto-id2688131\">An electron has a velocity [latex]v=0\\text{.}\\text{990}c[\/latex]. (a) Calculate the kinetic energy in MeV of the electron. (b) Compare this with the classical value for kinetic energy at this velocity. (The mass of an electron is [latex]9\\text{.}\\text{11}\u00d7{\\text{10}}^{-\\text{31}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex].)<\/p>\n<p id=\"import-auto-id3164659\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1639162\">The expression for relativistic kinetic energy is always correct, but for (a) it must be used since the velocity is highly relativistic (close to [latex]c[\/latex]). First, we will calculate the relativistic factor [latex]\\gamma [\/latex], and then use it to determine the relativistic kinetic energy. For (b), we will calculate the classical kinetic energy (which would be close to the relativistic value if [latex]v[\/latex] were less than a few percent of [latex]c[\/latex]) and see that it is not the same.<\/p>\n<p id=\"import-auto-id2837815\"><strong>Solution for (a) <\/strong><\/p>\n<ol id=\"fs-id1574907\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2546385\">Identify the knowns. [latex]v=0\\text{.}\\text{990}c[\/latex]; [latex]m=9\\text{.}\\text{11}\u00d7{\\text{10}}^{-\\text{31}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]<\/li>\n<li id=\"import-auto-id2597542\">Identify the unknown. [latex]{\\text{KE}}_{\\text{rel}}[\/latex]<\/li>\n<li id=\"import-auto-id3081018\">Choose the appropriate equation. [latex]{\\text{KE}}_{\\text{rel}}=\\left(\\gamma -1\\right){\\mathrm{mc}}^{2}[\/latex]<\/li>\n<li id=\"import-auto-id2595330\">Plug the knowns into the equation.\n<p id=\"import-auto-id1489564\">First calculate [latex]\\gamma [\/latex]. We will carry extra digits because this is an intermediate calculation.<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}\\gamma &amp; =&amp; \\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}\\\\ &amp; =&amp; \\frac{1}{\\sqrt{1-\\frac{\\left(0\\text{.}\\text{990}\\text{c}{\\right)}^{2}}{{c}^{2}}}}\\\\ &amp; =&amp; \\frac{1}{\\sqrt{1-\\left(0\\text{.}\\text{990}{\\right)}^{2}}}\\\\ &amp; =&amp; 7\\text{.}\\text{0888}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1535841\">Next, we use this value to calculate the kinetic energy.<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}{\\text{KE}}_{\\text{rel}}&amp; =&amp; \\left(\\gamma -1\\right){\\mathrm{mc}}^{2}\\\\ &amp; =&amp; \\left(\\text{7.0888}-1\\right)\\left(9.11\u00d7{\\text{10}}^{\u2013\\text{31}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\right)\\left(3.00\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}{\\right)}^{2}\\\\ &amp; =&amp; 4.99\u00d7{\\text{10}}^{\\text{\u201313}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\end{array}[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2003431\">Convert units.\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}{\\text{KE}}_{\\text{rel}}&amp; =&amp; \\left(4.99\u00d7{\\text{10}}^{\\text{\u201313}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\right)\\left(\\frac{\\text{1 MeV}}{1.60\u00d7{\\text{10}}^{\u2013\\text{13}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}}\\right)\\\\ &amp; =&amp; \\text{3.12 MeV}\\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p><strong>Solution for (b) <\/strong><\/p>\n<ol id=\"fs-id2584774\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2609096\">List the knowns. [latex]v=0\\text{.}\\text{990}c[\/latex]; [latex]m=9\\text{.}\\text{11}\u00d7{\\text{10}}^{-\\text{31}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]<\/li>\n<li id=\"import-auto-id2660805\">List the unknown. [latex]{\\text{KE}}_{\\text{class}}[\/latex]<\/li>\n<li id=\"import-auto-id1591594\">Choose the appropriate equation. [latex]{\\text{KE}}_{\\text{class}}=\\frac{1}{2}{\\mathrm{mv}}^{2}[\/latex]<\/li>\n<li id=\"import-auto-id3143593\">Plug the knowns into the equation.\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}{\\text{KE}}_{\\text{class}}&amp; =&amp; \\frac{1}{2}{\\mathrm{mv}}^{2}\\\\ &amp; =&amp; \\frac{1}{2}\\left(9.00\u00d7{\\text{10}}^{\u2013\\text{31}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\right)\\left(0.990{\\right)}^{2}\\left(3.00\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}{\\right)}^{2}\\\\ &amp; =&amp; 4.02\u00d7{\\text{10}}^{\u2013\\text{14}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\end{array}[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1505723\">Convert units.\n<div data-type=\"equation\" class=\"equation\" id=\"eip-719\">[latex]\\begin{array}{lll}{\\text{KE}}_{\\text{class}}&amp; =&amp; 4.02\u00d7{\\text{10}}^{\u2013\\text{14}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\left(\\frac{\\text{1 MeV}}{1.60\u00d7{\\text{10}}^{\u2013\\text{13}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}}\\right)\\\\ &amp; =&amp; \\text{0.251 MeV}\\end{array}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id2771680\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1523780\">As might be expected, since the velocity is 99.0% of the speed of light, the classical kinetic energy is significantly off from the correct relativistic value. Note also that the classical value is much smaller than the relativistic value. In fact, [latex]{\\text{KE}}_{\\text{rel}}{\\text{\/KE}}_{\\text{class}}=\\text{12}\\text{.}4[\/latex] here. This is some indication of how difficult it is to get a mass moving close to the speed of light. Much more energy is required than predicted classically. Some people interpret this extra energy as going into increasing the mass of the system, but, as discussed in <a href=\"\/contents\/cb474108-fe67-4904-bdbe-441b271ec10f\">Relativistic Momentum<\/a>, this cannot be verified unambiguously. What is certain is that ever-increasing amounts of energy are needed to get the velocity of a mass a little closer to that of light. An energy of 3 MeV is a very small amount for an electron, and it can be achieved with present-day particle accelerators. SLAC, for example, can accelerate electrons to over [latex]\\text{50}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{eV}=\\text{50,000 MeV}[\/latex].<\/p>\n<p id=\"import-auto-id2910636\">Is there any point in getting [latex]v[\/latex] a little closer to c than 99.0% or 99.9%? The answer is yes. We learn a great deal by doing this. The energy that goes into a high-velocity mass can be converted to any other form, including into entirely new masses. (See <a href=\"#import-auto-id3098026\" class=\"autogenerated-content\">(Figure)<\/a>.) Most of what we know about the substructure of matter and the collection of exotic short-lived particles in nature has been learned this way. Particles are accelerated to extremely relativistic energies and made to collide with other particles, producing totally new species of particles. Patterns in the characteristics of these previously unknown particles hint at a basic substructure for all matter. These particles and some of their characteristics will be covered in <a href=\"\/contents\/43a84abb-7daf-467e-ae21-b1b74baab670\">Particle Physics<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3098026\">\n<div class=\"bc-figcaption figcaption\">The Fermi National Accelerator Laboratory, near Batavia, Illinois, was a subatomic particle collider that accelerated protons and antiprotons to attain energies up to 1 Tev (a trillion electronvolts). The circular ponds near the rings were built to dissipate waste heat. This accelerator was shut down in September 2011. (credit: Fermilab, Reidar Hahn)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3098027\" data-alt=\"An aerial view of the Fermi National Accelerator Laboratory. The accelerator has two large, ring shaped structures. There are circular ponds near the rings.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_29_06_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"An aerial view of the Fermi National Accelerator Laboratory. The accelerator has two large, ring shaped structures. There are circular ponds near the rings.\" width=\"250\"><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2573402\">\n<h1 data-type=\"title\">Relativistic Energy and Momentum<\/h1>\n<p id=\"import-auto-id2724159\">We know classically that kinetic energy and momentum are related to each other, since<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\text{KE}}_{\\text{class}}=\\frac{{p}^{2}}{2m}=\\frac{\\left(\\mathrm{mv}{\\right)}^{2}}{2m}=\\frac{1}{2}{\\mathrm{mv}}^{2}.[\/latex]<\/div>\n<p id=\"import-auto-id2603540\">Relativistically, we can obtain a relationship between energy and momentum by algebraically manipulating their definitions. This produces<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{E}^{2}=\\left(\\text{pc}{\\right)}^{2}+\\left({\\text{mc}}^{2}{\\right)}^{2},[\/latex]<\/div>\n<p id=\"import-auto-id2634770\">where [latex]E[\/latex] is the relativistic total energy and [latex]p[\/latex] is the relativistic momentum. This relationship between relativistic energy and relativistic momentum is more complicated than the classical, but we can gain some interesting new insights by examining it. First, total energy is related to momentum and rest mass. At rest, momentum is zero, and the equation gives the total energy to be the rest energy [latex]{\\mathrm{mc}}^{2}[\/latex] (so this equation is consistent with the discussion of rest energy above). However, as the mass is accelerated, its momentum [latex]p[\/latex] increases, thus increasing the total energy. At sufficiently high velocities, the rest energy term [latex]\\left({\\mathrm{mc}}^{2}{\\right)}^{2}[\/latex] becomes negligible compared with the momentum term <\/p>\n<p>[latex]\\left(\\mathrm{pc}{\\right)}^{2}[\/latex]; thus, <\/p>\n<p>[latex]E=\\mathrm{pc}[\/latex] at extremely relativistic velocities.<\/p>\n<p id=\"import-auto-id1357067\">If we consider momentum [latex]p[\/latex] to be distinct from mass, we can determine the implications of the equation [latex]{E}^{2}=\\left(\\text{pc}{\\right)}^{2}+\\left({\\text{mc}}^{2}{\\right)}^{2},[\/latex]  for a particle that has no mass. If we take [latex]m[\/latex] to be zero in this equation, then [latex]E=\\text{pc}[\/latex], or [latex]p=E\/c[\/latex]. Massless particles have this momentum. There are several massless particles found in nature, including photons (these are quanta of electromagnetic radiation). Another implication is that a massless particle must travel at speed [latex]c[\/latex] and only at speed [latex]c[\/latex]. While it is beyond the scope of this text to examine the relationship in the equation [latex]{E}^{2}=\\left(\\text{pc}{\\right)}^{2}+\\left({\\text{mc}}^{2}{\\right)}^{2},[\/latex] in detail, we can see that the relationship has important implications in special relativity.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2607341\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Problem-Solving Strategies for Relativity<\/div>\n<ol id=\"fs-id2677927\" data-number-style=\"arabic\" class=\"stepwise\">\n<li id=\"import-auto-id3067421\"><em data-effect=\"italics\"> Examine the situation to determine that it is necessary to use relativity<\/em>. Relativistic effects are related to [latex]\\gamma =\\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}[\/latex], the quantitative relativistic factor. If [latex]\\gamma [\/latex] is very close to 1, then relativistic effects are small and differ very little from the usually easier classical calculations.<\/li>\n<li id=\"import-auto-id1675588\"><em data-effect=\"italics\"> Identify exactly what needs to be determined in the problem (identify the unknowns).<\/em><\/li>\n<li id=\"import-auto-id1364684\"><em data-effect=\"italics\"> Make a list of what is given or can be inferred from the problem as stated (identify the knowns).<\/em> Look in particular for information on relative velocity [latex]v[\/latex].<\/li>\n<li id=\"import-auto-id2608837\"><em data-effect=\"italics\"> Make certain you understand the conceptual aspects of the problem before making any calculations.<\/em> Decide, for example, which observer sees time dilated or length contracted before plugging into equations. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.<\/li>\n<li id=\"import-auto-id2515324\"><em data-effect=\"italics\"> Determine the primary type of calculation to be done to find the unknowns identified above.<\/em> You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.<\/li>\n<li id=\"import-auto-id2651965\"><em data-effect=\"italics\"> Do not round off during the calculation.<\/em> As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem, but do not use a rounded number in a subsequent calculation.<\/li>\n<li id=\"import-auto-id1357009\"><em data-effect=\"italics\"> Check the answer to see if it is reasonable: Does it make sense?<\/em> This may be more difficult for relativity, since we do not encounter it directly. But you can look for velocities greater than [latex]c[\/latex] or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"eip-id1537860\">\n<p id=\"eip-id1537861\">A photon decays into an electron-positron pair. What is the kinetic energy of the electron if its speed is  [latex]0.992c[\/latex]?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id1537883\" data-label=\"\">\n<div data-type=\"title\">Answer<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1537888\">[latex]\\begin{array}{lll}{\\text{KE}}_{\\text{rel}}&amp; =&amp; \\left(\\gamma -1\\right){\\mathrm{mc}}^{2}=\\left(\\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}-1\\right){\\mathrm{mc}}^{2}\\\\ &amp; =&amp; \\left(\\frac{1}{\\sqrt{1-\\frac{\\left(\\text{0.992}c{\\right)}^{2}}{{c}^{2}}}}-1\\right)\\left(\\text{9.11}\u00d7{\\text{10}}^{-\\text{31}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\right)\\left(\\text{3.00}\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}{\\right)}^{2}=\\text{5.67}\u00d7{\\text{10}}^{-\\text{13}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3107783\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2999106\">\n<li id=\"import-auto-id2747191\">Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.<\/li>\n<li id=\"import-auto-id2944323\">Total Energy is defined as:<br>\n[latex]E={\\mathrm{\\gamma mc}}^{2}[\/latex], where \n<p>[latex]\\gamma =\\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}[\/latex].<\/p><\/li>\n<li id=\"import-auto-id2821670\">Rest energy is<br>\n[latex]{E}_{0}={\\mathrm{mc}}^{2}[\/latex],  meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.<\/li>\n<li id=\"import-auto-id2972722\">We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy.<\/li>\n<li id=\"import-auto-id3143536\">The relativistic work-energy theorem is [latex]{W}_{\\text{net}}=E-{E}_{0}=\\gamma {\\mathrm{mc}}^{2}-{\\mathrm{mc}}^{2}=\\left(\\gamma -1\\right){\\mathrm{mc}}^{2}[\/latex].<\/li>\n<li>Relativistically,<br>\n[latex]{W}_{\\text{net}}={\\text{KE}}_{\\text{rel}}[\/latex] ,<br>\nwhere  [latex]{\\text{KE}}_{\\text{rel}}[\/latex] is the relativistic kinetic energy.<\/li>\n<li id=\"import-auto-id2768271\">Relativistic kinetic energy is [latex]{\\text{KE}}_{\\text{rel}}=\\left(\\gamma -1\\right){\\mathrm{mc}}^{2}[\/latex], where [latex]\\gamma =\\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}[\/latex]. At low velocities, relativistic kinetic energy reduces to classical kinetic energy.<\/li>\n<li id=\"import-auto-id3103866\"><strong>No object with mass can attain the speed of light<\/strong> because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.<\/li>\n<li id=\"import-auto-id2072642\">The equation [latex]{E}^{2}=\\left(\\mathrm{pc}{\\right)}^{2}+\\left({\\mathrm{mc}}^{2}{\\right)}^{2}[\/latex] relates the relativistic total energy\n<p>[latex]E[\/latex] and the relativistic momentum<br>\n[latex]p[\/latex]. At extremely high velocities, the rest energy<br>\n[latex]{\\mathrm{mc}}^{2}[\/latex] becomes negligible, and<br>\n[latex]E=\\mathrm{pc}[\/latex].<\/p><\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id3113705\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1563904\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1546041\">\n<p id=\"import-auto-id2762569\">How are the classical laws of conservation of energy and conservation of mass modified by modern relativity?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2962135\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2962136\">\n<p id=\"import-auto-id2712204\">What happens to the mass of water in a pot when it cools, assuming no molecules escape or are added? Is this observable in practice? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2985598\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2527016\">\n<p id=\"import-auto-id1256380\">Consider a thought experiment. You place an expanded balloon of air on weighing scales outside in the early morning. The balloon stays on the scales and you are able to measure changes in its mass. Does the mass of the balloon change as the day progresses? Discuss the difficulties in carrying out this experiment.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2768272\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3459817\">\n<p id=\"import-auto-id1550823\">The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2634587\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2634588\">\n<p id=\"import-auto-id2634720\">We know that the velocity of an object with mass has an upper limit of [latex]c[\/latex]. Is there an upper limit on its momentum? Its energy? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2834533\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2834534\">\n<p id=\"import-auto-id2602233\">Given the fact that light travels at [latex]c[\/latex], can it have mass? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2633453\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2633454\">\n<p id=\"import-auto-id2963402\">If you use an Earth-based telescope to project a laser beam onto the Moon, you can move the spot across the Moon\u2019s surface at a velocity greater than the speed of light. Does this violate modern relativity? (Note that light is being sent from the Earth to the Moon, not across the surface of the Moon.)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id3164432\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3084928\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2944361\">\n<p id=\"import-auto-id1999778\">What is the rest energy of an electron, given its mass is [latex]9\\text{.}\\text{11}\u00d7{\\text{10}}^{-\\text{31}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]? Give your answer in joules and MeV.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1959372\">\n<p id=\"import-auto-id1433005\">[latex]8.20\u00d7{\\text{10}}^{-\\text{14}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex]<\/p>\n<p id=\"import-auto-id2724228\">0.512 MeV<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2721624\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2721625\">\n<p id=\"import-auto-id2573635\">Find the rest energy in joules and MeV of a proton, given its mass is [latex]1\\text{.}\\text{67}\u00d7{\\text{10}}^{-\\text{27}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2681831\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2573985\">\n<p id=\"import-auto-id2644941\">If the rest energies of a proton and a neutron (the two constituents of nuclei) are 938.3 and 939.6 MeV respectively, what is the difference in their masses in kilograms?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1985675\">\n<p id=\"import-auto-id2675917\">[latex]2\\text{.}3\u00d7{\\text{10}}^{-\\text{30}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2720780\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2720781\">\n<p id=\"import-auto-id2736047\">The Big Bang that began the universe is estimated to have released [latex]{10}^{\\text{68}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex] of energy. How many stars could half this energy create, assuming the average star\u2019s mass is [latex]4\\text{.}\\text{00}\u00d7{\\text{10}}^{\\text{30}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2512400\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1362793\">\n<p id=\"import-auto-id3138939\">A supernova explosion of a [latex]2\\text{.}\\text{00}\u00d7{\\text{10}}^{\\text{31}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex] star produces [latex]1\\text{.}\\text{00}\u00d7{\\text{10}}^{\\text{44}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex] of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio [latex]\\Delta m\/m[\/latex] of mass destroyed to the original mass of the star?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1605062\">\n<p id=\"import-auto-id1565184\">(a) [latex]1\\text{.}\\text{11}\u00d7{\\text{10}}^{\\text{27}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]\n    <\/p>\n<p id=\"import-auto-id1563498\">(b) [latex]5\\text{.}\\text{56}\u00d7{\\text{10}}^{-5}[\/latex]\n    <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2572932\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2572933\">\n<p id=\"import-auto-id2912658\">(a) Using data from <a href=\"\/contents\/c5adb012-fbab-41a7-b5b2-4644984097fc#import-auto-id2866785\" class=\"autogenerated-content\">(Figure)<\/a>, calculate the mass converted to energy by the fission of 1.00 kg of uranium. (b) What is the ratio of mass destroyed to the original mass, [latex]\\Delta m\/m[\/latex]?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1539307\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1539308\">\n<p id=\"import-auto-id1363138\">(a) Using data from <a href=\"\/contents\/c5adb012-fbab-41a7-b5b2-4644984097fc#import-auto-id2866785\" class=\"autogenerated-content\">(Figure)<\/a>, calculate the amount of mass converted to energy by the fusion of 1.00 kg of hydrogen. (b) What is the ratio of mass destroyed to the original mass, [latex]\\Delta m\/m[\/latex]? (c) How does this compare with [latex]\\Delta m\/m[\/latex] for the fission of 1.00 kg of uranium?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id3084938\">[latex]7\\text{.}1\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]<\/p>\n<p id=\"import-auto-id2631704\">[latex]7\\text{.}1\u00d7{\\text{10}}^{-3}[\/latex]<\/p>\n<p id=\"import-auto-id3143597\">The ratio is greater for hydrogen.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1780374\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1780375\">\n<p id=\"import-auto-id1612975\">There is approximately [latex]{10}^{\\text{34}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex] of energy available from fusion of hydrogen in the world\u2019s oceans. (a) If [latex]{10}^{\\text{33}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex] of this energy were utilized, what would be the decrease in mass of the oceans? (b) How great a volume of water does this correspond to? (c) Comment on whether this is a significant fraction of the total mass of the oceans.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1350397\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1350398\">\n<p id=\"import-auto-id2575969\">A muon has a rest mass energy of 105.7 MeV, and it decays into an electron and a massless particle. (a) If all the lost mass is converted into the electron\u2019s kinetic energy, find [latex]\\gamma [\/latex] for the electron. (b) What is the electron\u2019s velocity?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2560231\">\n<p id=\"import-auto-id2003279\">208<\/p>\n<p id=\"import-auto-id1956060\">[latex]0.999988c[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2702554\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2702555\">\n<p id=\"import-auto-id2607634\">A [latex]\\pi [\/latex]-meson is a particle that decays into a muon and a massless particle. The [latex]\\pi [\/latex]-meson has a rest mass energy of 139.6 MeV, and the muon has a rest mass energy of 105.7 MeV. Suppose the [latex]\\pi [\/latex]-meson is at rest and all of the missing mass goes into the muon\u2019s kinetic energy. How fast will the muon move?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3358455\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3358456\">\n<p id=\"import-auto-id2651913\">(a) Calculate the relativistic kinetic energy of a 1000-kg car moving at 30.0 m\/s if the speed of light were only 45.0 m\/s. (b) Find the ratio of the relativistic kinetic energy to classical.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2773029\">\n<p id=\"import-auto-id1440827\">[latex]6.92\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex]<\/p>\n<p id=\"import-auto-id1573461\">1.54<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2753673\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2753674\">\n<p id=\"import-auto-id2594708\">Alpha decay is nuclear decay in which a helium nucleus is emitted. If the helium nucleus has a mass of [latex]6\\text{.}\\text{80}\u00d7{\\text{10}}^{-\\text{27}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex] and is given 5.00 MeV of kinetic energy, what is its velocity?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3195968\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3195969\">\n<p id=\"import-auto-id3204662\">(a) Beta decay is nuclear decay in which an electron is emitted. If the electron is given 0.750 MeV of kinetic energy, what is its velocity? (b) Comment on how the high velocity is consistent with the kinetic energy as it compares to the rest mass energy of the electron.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3137588\">\n<p>(a) [latex]0\\text{.}\\text{914}c[\/latex]\n    <\/p>\n<p id=\"import-auto-id2783066\">(b) The<br>\nrest<br>\nmass<br>\nenergy<br>\nof<br>\nan<br>\nelectron<br>\nis<br>\n0.511<br>\nMeV,<br>\nso<br>\nthe<br>\nkinetic<br>\nenergy<br>\nis<br>\napproximately<br>\n150%<br>\nof<br>\nthe<br>\nrest<br>\nmass<br>\nenergy.<br>\nThe<br>\nelectron<br>\nshould<br>\nbe<br>\ntraveling<br>\nclose<br>\nto<br>\nthe<br>\nspeed<br>\nof<br>\nlight.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2600562\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2600563\">\n<p id=\"import-auto-id2592460\">A positron is an antimatter version of the electron, having exactly the same mass. When a positron and an electron meet, they annihilate, converting all of their mass into energy. (a) Find the energy released, assuming negligible kinetic energy before the annihilation. (b) If this energy is given to a proton in the form of kinetic energy, what is its velocity? (c) If this energy is given to another electron in the form of kinetic energy, what is its velocity?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1284916\">\n<p id=\"import-auto-id2595222\">What is the kinetic energy in MeV of a [latex]\\pi [\/latex]-meson that lives [latex]1\\text{.}\\text{40}\u00d7{10}^{-\\text{16}}\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex] as measured in the laboratory, and [latex]0\\text{.}\\text{840}\u00d7{10}^{-\\text{16}}\\phantom{\\rule{0.25em}{0ex}}\\text{s}[\/latex] when at rest relative to an observer, given that its rest energy is 135 MeV?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2723222\">\n<p id=\"import-auto-id2724319\">90.0 MeV<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2725006\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2725008\">\n<p id=\"import-auto-id3101739\">Find the kinetic energy in MeV of a neutron with a measured life span of 2065 s, given its rest energy is 939.6 MeV, and rest life span is 900s.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2500966\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2500967\">\n<p id=\"import-auto-id1656559\">(a) Show that [latex]\\left(\\mathrm{pc}{\\right)}^{2}\/\\left({\\mathrm{mc}}^{2}{\\right)}^{2}={\\gamma }^{2}-1[\/latex]. This means that at large velocities [latex]\\mathrm{pc}\\text{&gt;&gt;}{\\mathrm{mc}}^{2}[\/latex]. (b) Is [latex]E\\approx \\text{pc}[\/latex] when [latex]\\gamma =\\text{30}\\text{.}0[\/latex], as for the astronaut discussed in the twin paradox?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3164547\">\n<p id=\"eip-id1432763\">(a)     [latex]\\begin{array}{l}{E}^{2}={p}^{2}{c}^{2}+{m}^{2}{c}^{4}={\\gamma }^{2}{m}^{2}{c}^{4},\\phantom{\\rule{0.25em}{0ex}}\\text{so that}\\\\ {p}^{2}{c}^{2}=\\left({\\gamma }^{2}-1\\right){m}^{2}{c}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{, and therefore}\\\\ \\phantom{\\rule{0.25em}{0ex}}\\frac{{\\left(\\text{pc}\\right)}^{2}}{{\\left({\\mathit{\\text{mc}}}^{2}\\right)}^{2}}={\\gamma }^{2}-1\\end{array}[\/latex]<\/p>\n<p id=\"import-auto-id3087534\">(b) yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2834155\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2608664\">\n<p id=\"import-auto-id2509029\">One cosmic ray neutron has a velocity of [latex]0.250c[\/latex] relative to the Earth. (a) What is the neutron\u2019s total energy in MeV? (b) Find its momentum. (c) Is [latex]E\\approx \\text{pc}[\/latex] in this situation? Discuss in terms of the equation given in part (a) of the previous problem.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2629153\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2629154\">\n<p id=\"import-auto-id3088688\">What is [latex]\\gamma [\/latex] for a proton having a mass energy of 938.3 MeV accelerated through an effective potential of 1.0 TV (teravolt) at Fermilab outside Chicago?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2942320\">\n<p id=\"import-auto-id2009307\">[latex]1\\text{.}\\text{07}\u00d7{\\text{10}}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3164443\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3164444\">\n<p id=\"import-auto-id2676284\">(a) What is the effective accelerating potential for electrons at the Stanford Linear Accelerator, if [latex]\\gamma =1\\text{.}\\text{00}\u00d7{\\text{10}}^{5}[\/latex] for them? (b) What is their total energy (nearly the same as kinetic in this case) in GeV?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2911245\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2911246\">\n<p id=\"import-auto-id2502156\">(a) Using data from <a href=\"\/contents\/c5adb012-fbab-41a7-b5b2-4644984097fc#import-auto-id2866785\" class=\"autogenerated-content\">(Figure)<\/a>, find the mass destroyed when the energy in a barrel of crude oil is released. (b) Given these barrels contain 200 liters and assuming the density of crude oil is [latex]750 kg{\\text{\/m}}^{3}[\/latex], what is the ratio of mass destroyed to original mass, [latex]\\Delta m\/m[\/latex]?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3165931\">\n<p id=\"import-auto-id2923850\">[latex]6\\text{.}\\text{56}\u00d7{\\text{10}}^{-8}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]<\/p>\n<p id=\"import-auto-id2977369\">[latex]4.37\u00d7{\\text{10}}^{-\\text{10}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3259500\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3259501\">\n<p id=\"import-auto-id2746932\">(a) Calculate the energy released by the destruction of 1.00 kg of mass. (b) How many kilograms could be lifted to a 10.0 km height by this amount of energy?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2656682\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2656683\">\n<p id=\"import-auto-id1415466\">A Van de Graaff accelerator utilizes a 50.0 MV potential difference to accelerate charged particles such as protons. (a) What is the velocity of a proton accelerated by such a potential? (b) An electron?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3040526\">\n<p id=\"import-auto-id1415472\">[latex]0.314c[\/latex]<\/p>\n<p id=\"eip-id1461415\">[latex]0.99995c[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3143371\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3143372\">\n<p id=\"import-auto-id3371246\">Suppose you use an average of [latex]500 kW\u00b7h[\/latex]  of electric energy per month in your home. (a) How long would 1.00 g of mass converted to electric energy with an efficiency of 38.0% last you? (b) How many homes could be supplied at the [latex]500 kW\u00b7h[\/latex] per month rate for one year by the energy from the described mass conversion?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1946210\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3158295\">\n<p id=\"import-auto-id1356748\">(a) A nuclear power plant converts energy from nuclear fission into electricity with an efficiency of 35.0%. How much mass is destroyed in one year to produce a continuous 1000 MW of electric power? (b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is [latex]{10}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2836488\">\n<p id=\"import-auto-id2657369\">(a) 1.00 kg<\/p>\n<p id=\"eip-id2744805\">(b) This much mass would be measurable, but probably not observable just by looking because it is 0.01% of the total mass.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3159094\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3159095\">\n<p>Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket\u2019s mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of [latex]1.00\u00d7{10}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex] (100 tons), what total yield nuclear explosion in tons of TNT is needed?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2693263\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2693264\">\n<p id=\"import-auto-id3152001\">The Sun produces energy at a rate of [latex]4\\text{.}\\text{00}\u00d7{\\text{10}}^{\\text{26}}[\/latex] W by the fusion of hydrogen. (a) How many kilograms of hydrogen undergo fusion each second? (b) If the Sun is 90.0% hydrogen and half of this can undergo fusion before the Sun changes character, how long could it produce energy at its current rate? (c) How many kilograms of mass is the Sun losing per second? (d) What fraction of its mass will it have lost in the time found in part (b)?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2602531\">\n<p id=\"import-auto-id3084632\">(a) [latex]6\\text{.}3\u00d7{\\text{10}}^{\\text{11}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg\/s}[\/latex]\n    <\/p>\n<p id=\"import-auto-id3087172\">(b) [latex]4\\text{.}5\u00d7{\\text{10}}^{\\text{10}}\\phantom{\\rule{0.25em}{0ex}}\\text{y}[\/latex]\n    <\/p>\n<p id=\"import-auto-id2761625\">(c) [latex]4\\text{.}\\text{44}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]\n    <\/p>\n<p id=\"import-auto-id2693455\">(d) 0.32%<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2791387\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2791388\">\n<p id=\"eip-id3377332\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id3109827\">A proton has a mass of [latex]1\\text{.}\\text{67}\u00d7{\\text{10}}^{-\\text{27}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]. A physicist measures the proton\u2019s total energy to be 50.0 MeV. (a) What is the proton\u2019s kinetic energy? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1936682\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1936683\">\n<p id=\"import-auto-id3067560\"><strong>Construct Your Own Problem<\/strong><\/p>\n<p id=\"eip-id2312930\">Consider a highly relativistic particle. Discuss what is meant by the term \u201chighly relativistic.\u201d (Note that, in part, it means that the particle cannot be massless.) Construct a problem in which you calculate the wavelength of such a particle and show that it is very nearly the same as the wavelength of a massless particle, such as a photon, with the same energy. Among the things to be considered are the rest energy of the particle (it should be a known particle) and its total energy, which should be large compared to its rest energy.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2929431\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2929432\">\n<p id=\"import-auto-id2657479\"><strong>Construct Your Own Problem<\/strong><\/p>\n<p id=\"eip-id1632086\">Consider an astronaut traveling to another star at a relativistic velocity. Construct a problem in which you calculate the time for the trip as observed on the Earth and as observed by the astronaut. Also calculate the amount of mass that must be converted to energy to get the astronaut and ship to the velocity travelled. Among the things to be considered are the distance to the star, the velocity, and the mass of the astronaut and ship. Unless your instructor directs you otherwise, do not include any energy given to other masses, such as rocket propellants.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1364291\">\n<dt>total energy<\/dt>\n<dd id=\"fs-id1360334\"> defined as [latex]E={\\mathrm{\\gamma mc}}^{2}[\/latex], where [latex]\\gamma =\\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1781799\">\n<dt>rest energy<\/dt>\n<dd id=\"fs-id2677386\">the energy stored in an object at rest: [latex]{E}_{0}={\\mathrm{mc}}^{2}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3167814\">\n<dt>relativistic kinetic energy<\/dt>\n<dd id=\"fs-id1359660\">the kinetic energy of an object moving at relativistic speeds: [latex]{\\text{KE}}_{\\text{rel}}=\\left(\\gamma -1\\right){\\mathrm{mc}}^{2}[\/latex], where [latex]\\gamma =\\frac{1}{\\sqrt{1-\\frac{{v}^{2}}{{c}^{2}}}}[\/latex]<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Compute total energy of a relativistic object.<\/li>\n<li>Compute the kinetic energy of a relativistic object.<\/li>\n<li>Describe rest energy, and explain how it can be converted to other forms.<\/li>\n<li>Explain why massive particles cannot reach C.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id882829\">\n<div class=\"bc-figcaption figcaption\">The National Spherical Torus Experiment (NSTX) has a fusion reactor in which hydrogen isotopes undergo fusion to produce helium. In this process, a relatively small mass of fuel is converted into a large amount of energy. (credit: Princeton Plasma Physics Laboratory)<\/div>\n<p><span data-type=\"media\" data-alt=\"This photo shows the outside of the fusion reactor of the National Spherical Torus Experiment at the Princeton Plasma Physics Laboratory. The reactor, which sits in a large room, is connected to numerous tubes and instruments.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_29_06_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"This photo shows the outside of the fusion reactor of the National Spherical Torus Experiment at the Princeton Plasma Physics Laboratory. The reactor, which sits in a large room, is connected to numerous tubes and instruments.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1985448\">A tokamak is a form of experimental fusion reactor, which can change mass to energy. Accomplishing this requires an understanding of relativistic energy. Nuclear reactors are proof of the conservation of relativistic energy.<\/p>\n<p id=\"import-auto-id1284621\">Conservation of energy is one of the most important laws in physics. Not only does energy have many important forms, but each form can be converted to any other. We know that classically the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is altered to include the possibility of mass changing to energy, as in the reactions that occur within a nuclear reactor. Relativistic energy is intentionally defined so that it will be conserved in all inertial frames, just as is the case for relativistic momentum. As a consequence, we learn that several fundamental quantities are related in ways not known in classical physics. All of these relationships are verified by experiment and have fundamental consequences. The altered definition of energy contains some of the most fundamental and spectacular new insights into nature found in recent history.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2633718\">\n<h1 data-type=\"title\">Total Energy and Rest Energy<\/h1>\n<p id=\"import-auto-id3137655\">The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define energy to include a relativistic factor.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2636850\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Total Energy<\/div>\n<p id=\"import-auto-id1309522\"><span data-type=\"term\" id=\"import-auto-id2601670\">Total energy <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is defined to be<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4765a8449381deb7e127f75ff9b5c034_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id3163669\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> is mass, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the speed of light, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7542a73db72a1290e0d44df531398e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"85\" style=\"vertical-align: -21px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the velocity of the mass relative to an observer. There are many aspects of the total energy <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> that we will discuss\u2014among them are how kinetic and potential energies are included in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, and how <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is related to relativistic momentum. But first, note that at rest, total energy is not zero. Rather, when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-67b297dfc2c97b2e282dc2716c601577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/>, we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2da4984511012f81e5c0971541c66695_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/>, and an object has rest energy.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3099697\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Rest Energy<\/div>\n<p id=\"import-auto-id2644699\"><span data-type=\"term\" id=\"import-auto-id2678807\">Rest energy<\/span> is\n    <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-594\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9f6721dd8de2925d0cf5fe30e63d68cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<p id=\"import-auto-id2994648\">This is the correct form of Einstein\u2019s most famous equation, which for the first time showed that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. The implications of these first two equations regarding relativistic energy are so broad that they were not completely recognized for some years after Einstein published them in 1907, nor was the experimental proof that they are correct widely recognized at first. Einstein, it should be noted, did understand and describe the meanings and implications of his theory.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1516569\">\n<div data-type=\"title\" class=\"title\">Calculating Rest Energy: Rest Energy is Very Large<\/div>\n<p id=\"import-auto-id1599949\">Calculate the rest energy of a 1.00-g mass.<\/p>\n<p id=\"import-auto-id2010072\"><strong>Strategy<\/strong><\/p>\n<p>One gram is a small mass\u2014less than half the mass of a penny. We can multiply this mass, in SI units, by the speed of light squared to find the equivalent rest energy.<\/p>\n<p id=\"import-auto-id1637047\"><strong>Solution<\/strong><\/p>\n<ol id=\"fs-id3459859\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2720059\">Identify the knowns. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03be58cf6012ee689e3a357b3136a40a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -3px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7cf07b358dc5d50a61ba22ec17d1fb07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/li>\n<li id=\"import-auto-id1279713\">Identify the unknown. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6d784f5d4d4b89e37d7fdb08b2f36435_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/><\/li>\n<li id=\"import-auto-id2632083\">Choose the appropriate equation. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-51e9548d34219b8907c811a780cdd4f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -3px;\" \/><\/li>\n<li>Plug the knowns into the equation.\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-349970015efafc1f372ba1ada26e9a73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#38;&#32;&#61;&#38;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#57;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"346\" style=\"vertical-align: -17px;\" \/><\/div>\n<\/li>\n<li>Convert units.\n<p id=\"import-auto-id1583964\">Noting that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-adc2fba169cae2c4158f2e3975eaed34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#32;&#74;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"134\" style=\"vertical-align: -4px;\" \/>, we see the rest mass energy is<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f31dfc37747aee23c70889c918bcfc6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id2555003\"><strong>Discussion<\/strong><\/p>\n<p>This is an enormous amount of energy for a 1.00-g mass. We do not notice this energy, because it is generally not available. Rest energy is large because the speed of light <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is a large number and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-48c3fb47d1938015d9e5007068f8f09f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: 0px;\" \/> is a very large number, so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2ca930fd3e1c1a242725e5324be20573_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"30\" style=\"vertical-align: 0px;\" \/> is huge for any macroscopic mass. The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-60a1ab14ca62c4f05ce4c269c540859e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -1px;\" \/> rest mass energy for 1.00 g is about twice the energy released by the Hiroshima atomic bomb and about 10,000 times the kinetic energy of a large aircraft carrier. If a way can be found to convert rest mass energy into some other form (and all forms of energy can be converted into one another), then huge amounts of energy can be obtained from the destruction of mass.<\/p>\n<\/div>\n<p>Today, the practical applications of <em data-effect=\"italics\"><em data-effect=\"italics\">the conversion of mass into another form of energy<\/em><\/em>, such as in nuclear weapons and nuclear power plants, are well known. But examples also existed when Einstein first proposed the correct form of relativistic energy, and he did describe some of them. Nuclear radiation had been discovered in the previous decade, and it had been a mystery as to where its energy originated. The explanation was that, in certain nuclear processes, a small amount of mass is destroyed and energy is released and carried by nuclear radiation. But the amount of mass destroyed is so small that it is difficult to detect that any is missing. Although Einstein proposed this as the source of energy in the radioactive salts then being studied, it was many years before there was broad recognition that mass could be and, in fact, commonly is converted to energy. (See <a href=\"#import-auto-id2636680\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2636680\">\n<div class=\"bc-figcaption figcaption\">The Sun (a) and the Susquehanna Steam Electric Station (b) both convert mass into energy\u2014the Sun via nuclear fusion, the electric station via nuclear fission. (credits: (a) NASA\/Goddard Space Flight Center, Scientific Visualization Studio; (b) U.S. government) <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2005219\" data-alt=\"Part a of the figure shows a solar storm on the Sun. Part b of the figure shows the Susquehanna Steam Electric Station, which produces electricity by nuclear fission.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_29_06_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows a solar storm on the Sun. Part b of the figure shows the Susquehanna Steam Electric Station, which produces electricity by nuclear fission.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2836811\">Because of the relationship of rest energy to mass, we now consider mass to be a form of energy rather than something separate. There had not even been a hint of this prior to Einstein\u2019s work. Such conversion is now known to be the source of the Sun\u2019s energy, the energy of nuclear decay, and even the source of energy keeping Earth\u2019s interior hot.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2585048\">\n<h1 data-type=\"title\">Stored Energy and Potential Energy<\/h1>\n<p id=\"import-auto-id1579956\">What happens to energy stored in an object at rest, such as the energy put into a battery by charging it, or the energy stored in a toy gun\u2019s compressed spring? The energy input becomes part of the total energy of the object and, thus, increases its rest mass. All stored and potential energy becomes mass in a system. Why is it we don\u2019t ordinarily notice this? In fact, conservation of mass (meaning total mass is constant) was one of the great laws verified by 19th-century science. Why was it not noticed to be incorrect? The following example helps answer these questions.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1493235\">\n<div data-type=\"title\" class=\"title\">Calculating Rest Mass: A Small Mass Increase due to Energy Input<\/div>\n<p id=\"import-auto-id2784873\">A car battery is rated to be able to move 600 ampere-hours <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-015e06f3cdd6e6ef4e863c011c91762f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#65;&middot;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> of charge at 12.0 V. (a) Calculate the increase in rest mass of such a battery when it is taken from being fully depleted to being fully charged. (b) What percent increase is this, given the battery\u2019s mass is 20.0 kg?<\/p>\n<p id=\"import-auto-id3169070\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2721940\">In part (a), we first must find the energy stored in the battery, which equals what the battery can supply in the form of electrical potential energy. Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-98324ce6b1b9c6fdbdb97c5c8399abd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#101;&#99;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"93\" style=\"vertical-align: -4px;\" \/>, we have to calculate the charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fe5f7f5d3e616396be3c8a692220c1c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#65;&middot;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: 0px;\" \/>, which is the product of the current <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> and the time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>. We then multiply the result by 12.0 V. We can then calculate the battery\u2019s increase in mass using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ee7b16bd9e51e42ed7ef11fd1e5a2b4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#69;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#101;&#99;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"185\" style=\"vertical-align: -4px;\" \/>. Part (b) is a simple ratio converted to a percentage.<\/p>\n<p id=\"import-auto-id2976450\"><strong>Solution for (a) <\/strong><\/p>\n<ol id=\"fs-id2749129\" data-number-style=\"arabic\">\n<li id=\"import-auto-id1610514\">Identify the knowns. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e0de409a028266885bbadecec1e6e4d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;&#92;&#99;&#100;&#111;&#116;&#32;&#116;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#48;&#32;&#65;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"121\" style=\"vertical-align: -1px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8509f4ba2bbc130d4f9a8da95e3228b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"87\" style=\"vertical-align: -1px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7cf07b358dc5d50a61ba22ec17d1fb07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/li>\n<li id=\"import-auto-id2955803\">Identify the unknown. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8b54ad14e30444378952afb43fab8c57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/li>\n<li id=\"import-auto-id2731830\">Choose the appropriate equation. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e35ea2fe5a34d19211ffa8dbcc3df90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#101;&#99;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/li>\n<li id=\"import-auto-id2561455\">Rearrange the equation to solve for the unknown. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b9fc74fa8ffbb5f03dbcd445140fca3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#101;&#99;&#125;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"95\" style=\"vertical-align: -7px;\" \/><\/li>\n<li id=\"import-auto-id1310375\">Plug the knowns into the equation.\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a397efeaf4ffdb8e11f641ce872cbb01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#108;&#101;&#99;&#125;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#86;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#73;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#86;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#48;&#32;&#65;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#46;&#48;&#32;&#86;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"200\" style=\"vertical-align: -49px;\" \/><\/div>\n<p id=\"import-auto-id1327415\">Write amperes A as coulombs per second (C\/s), and convert hours to seconds.<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-89cfc0a30fd822913b2e70bb836ed74a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#48;&#32;&#67;&#47;&#115;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#48;&#48;&#32;&#115;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#104;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#46;&#48;&#32;&#74;&#47;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#46;&#49;&#54;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#46;&#48;&#32;&#74;&#47;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"82\" width=\"269\" style=\"vertical-align: -36px;\" \/><\/div>\n<p id=\"import-auto-id1048320\">Using the conversion <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b76db61cdf29acbd1e11e83391896afe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"135\" style=\"vertical-align: -4px;\" \/>, we can write the mass as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-36a057b16de000218d6d93343a1e3cdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#61;&#50;&#46;&#56;&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"155\" style=\"vertical-align: -3px;\" \/>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id923710\"><strong>Solution for (b) <\/strong><\/p>\n<ol id=\"fs-id1290816\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2990812\">Identify the knowns. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7fc49b8dbb138b2da6df792455f65f73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#61;&#50;&#46;&#56;&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"151\" style=\"vertical-align: -3px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6bf5bba72d42afa64aee61b29bb7e875_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&#32;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/li>\n<li id=\"import-auto-id2703315\">Identify the unknown. % change<\/li>\n<li id=\"import-auto-id2604060\">Choose the appropriate equation.\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#37;&#32;&#105;&#110;&#99;&#114;&#101;&#97;&#115;&#101;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#125;&#123;&#109;&#125;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#37;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\n\n*** Error message:\n&#70;&#105;&#108;&#101;&#32;&#101;&#110;&#100;&#101;&#100;&#32;&#119;&#104;&#105;&#108;&#101;&#32;&#115;&#99;&#97;&#110;&#110;&#105;&#110;&#103;&#32;&#117;&#115;&#101;&#32;&#111;&#102;&#32;&#92;&#116;&#101;&#120;&#116;&#64;&#46;\r\n&#69;&#109;&#101;&#114;&#103;&#101;&#110;&#99;&#121;&#32;&#115;&#116;&#111;&#112;&#46;\r\n\n<\/pre>\n<\/li>\n<li id=\"import-auto-id2637141\">Plug the knowns into the equation.\n<div data-type=\"equation\" class=\"equation\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#37;&#32;&#105;&#110;&#99;&#114;&#101;&#97;&#115;&#101;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#125;&#123;&#109;&#125;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#37;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#46;&#56;&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&#32;&#107;&#103;&#125;&#125;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#37;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#49;&#46;&#52;&#52;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#37;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#70;&#105;&#108;&#101;&#32;&#101;&#110;&#100;&#101;&#100;&#32;&#119;&#104;&#105;&#108;&#101;&#32;&#115;&#99;&#97;&#110;&#110;&#105;&#110;&#103;&#32;&#117;&#115;&#101;&#32;&#111;&#102;&#32;&#92;&#116;&#101;&#120;&#116;&#64;&#46;\r\n&#69;&#109;&#101;&#114;&#103;&#101;&#110;&#99;&#121;&#32;&#115;&#116;&#111;&#112;&#46;\r\n\n<\/pre>\n<\/div>\n<\/li>\n<\/ol>\n<p><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2719499\">Both the actual increase in mass and the percent increase are very small, since energy is divided by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-48c3fb47d1938015d9e5007068f8f09f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: 0px;\" \/>, a very large number. We would have to be able to measure the mass of the battery to a precision of a billionth of a percent, or 1 part in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d0fd5ddc40425f9ed08155f006979da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"30\" style=\"vertical-align: -1px;\" \/>, to notice this increase. It is no wonder that the mass variation is not readily observed. In fact, this change in mass is so small that we may question how you could verify it is real. The answer is found in nuclear processes in which the percentage of mass destroyed is large enough to be measured. The mass of the fuel of a nuclear reactor, for example, is measurably smaller when its energy has been used. In that case, stored energy has been released (converted mostly to heat and electricity) and the rest mass has decreased. This is also the case when you use the energy stored in a battery, except that the stored energy is much greater in nuclear processes, making the change in mass measurable in practice as well as in theory.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id817296\">\n<h1 data-type=\"title\">Kinetic Energy and the Ultimate Speed Limit<\/h1>\n<p id=\"import-auto-id704456\">Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-01ccd178a831af9bc21d30119eca4292_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#118;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/>. The relativistic expression for kinetic energy is obtained from the work-energy theorem. This theorem states that the net work on a system goes into kinetic energy. If our system starts from rest, then the work-energy theorem is<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e144afb2d8f5369a6ce932b349f3a0ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#87;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1667111\">Relativistically, at rest we have rest energy <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-51e9548d34219b8907c811a780cdd4f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -3px;\" \/>. The work increases this to the total energy <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d2a4cd7bb455360758a0664cbf9c012_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/>. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-888\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d86c2c13190604e0aa52b1f83be96a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#87;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#69;&#45;&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"348\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2676240\">Relativistically, we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1edf594fea68219e837a741ffb90580c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#87;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1611892\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Relativistic Kinetic Energy<\/div>\n<p><span data-type=\"term\" id=\"import-auto-id2638011\">Relativistic kinetic energy<\/span> is\n    <\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-81ae15f10056afd89e1a0cb2d8dff4fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"import-auto-id3156287\">When motionless, we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-67b297dfc2c97b2e282dc2716c601577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> and<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c1aaa7ff132c58d277c3b78ffae37335_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;&#61;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"123\" style=\"vertical-align: -21px;\" \/><\/div>\n<p id=\"import-auto-id2501781\">so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-493ef1a6fe5c594a84754a112bbc7b88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"75\" style=\"vertical-align: -4px;\" \/> at rest, as expected. But the expression for relativistic kinetic energy (such as total energy and rest energy) does not look much like the classical <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-01ccd178a831af9bc21d30119eca4292_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#118;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/>. To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfd339d0f13026ff7af56aa6f129380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> at low velocities gives<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6905d7fdcf12deeb7cc4cec5d78ba3a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"97\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id3067754\">A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In some cases, as in the limit of small velocity here, most terms are very small. Thus the expression derived for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfd339d0f13026ff7af56aa6f129380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> here is not exact, but it is a very accurate approximation. Thus, at low velocities,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8b84e1788c42d68a2095514e4aa213d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"97\" style=\"vertical-align: -7px;\" \/><\/div>\n<p>Entering this into the expression for relativistic kinetic energy gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-962\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-33dfd7b6428da000cc90389bee64052d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#118;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#108;&#97;&#115;&#115;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"292\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"import-auto-id1362957\">So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9902c6954e97dae1365b59fa328af844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#92;&#116;&#101;&#120;&#116;&#123;&#60;&#60;&#125;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<p id=\"import-auto-id3354394\">It is even more interesting to investigate what happens to kinetic energy when the velocity of an object approaches the speed of light. We know that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfd339d0f13026ff7af56aa6f129380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> becomes infinite as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> approaches <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, so that KE<sub>rel<\/sub> also becomes infinite as the velocity approaches the speed of light. (See <a href=\"#import-auto-id1745737\" class=\"autogenerated-content\">(Figure)<\/a>.) An infinite amount of work (and, hence, an infinite amount of energy input) is required to accelerate a mass to the speed of light.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2836695\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">The Speed of Light<\/div>\n<p id=\"import-auto-id1562844\"><strong>No object with mass can attain the speed of light.<\/strong><\/p>\n<\/div>\n<p id=\"import-auto-id2687489\">So the speed of light is the ultimate speed limit for any particle having mass. All of this is consistent with the fact that velocities less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> always add to less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>. Both the relativistic form for kinetic energy and the ultimate speed limit being <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> have been confirmed in detail in numerous experiments. No matter how much energy is put into accelerating a mass, its velocity can only approach\u2014not reach\u2014the speed of light.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1745737\">\n<div class=\"bc-figcaption figcaption\">This graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-55813247d68384087a68f48a446e885b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/> versus velocity shows how kinetic energy approaches infinity as velocity approaches the speed of light. It is thus not possible for an object having mass to reach the speed of light. Also shown is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-709e40e41f013af51396f99ac556037a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#108;&#97;&#115;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"53\" style=\"vertical-align: -4px;\" \/>, the classical kinetic energy, which is similar to relativistic kinetic energy at low velocities. Note that much more energy is required to reach high velocities than predicted classically.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3136349\" data-alt=\"In this figure a graph is shown on a coordinate system of axes. The x-axis is labeled as speed v (m\/s). On the x-axis, velocity of the object is shown in terms of the speed of light starting from zero at origin to c, where c is the speed of light. The y-axis is labeled as Kinetic Energy K E (J). On the y-axis, relativistic kinetic energy is shown starting from 0 at origin to 1.0. The graph K sub r e l of relativistic kinetic energy is concave up and moving upward along the vertical line at x equals c. This graph shows that relativistic kinetic energy approaches infinity as the velocity of an object approaches the speed of light. Also shown is that when the speed of the object is equal to the speed of light c the kinetic energy is known as classical kinetic energy, which is denoted as K E sub class.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_29_06_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"In this figure a graph is shown on a coordinate system of axes. The x-axis is labeled as speed v (m\/s). On the x-axis, velocity of the object is shown in terms of the speed of light starting from zero at origin to c, where c is the speed of light. The y-axis is labeled as Kinetic Energy K E (J). On the y-axis, relativistic kinetic energy is shown starting from 0 at origin to 1.0. The graph K sub r e l of relativistic kinetic energy is concave up and moving upward along the vertical line at x equals c. This graph shows that relativistic kinetic energy approaches infinity as the velocity of an object approaches the speed of light. Also shown is that when the speed of the object is equal to the speed of light c the kinetic energy is known as classical kinetic energy, which is denoted as K E sub class.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1553241\">\n<div data-type=\"title\" class=\"title\">Comparing Kinetic Energy: Relativistic Energy Versus Classical Kinetic Energy<\/div>\n<p id=\"import-auto-id2688131\">An electron has a velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0654f4aad251855943737c7e0d273cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#48;&#125;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"81\" style=\"vertical-align: 0px;\" \/>. (a) Calculate the kinetic energy in MeV of the electron. (b) Compare this with the classical value for kinetic energy at this velocity. (The mass of an electron is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7d6fa3fd511a85e751573877fa403984_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -3px;\" \/>.)<\/p>\n<p id=\"import-auto-id3164659\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1639162\">The expression for relativistic kinetic energy is always correct, but for (a) it must be used since the velocity is highly relativistic (close to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>). First, we will calculate the relativistic factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfd339d0f13026ff7af56aa6f129380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/>, and then use it to determine the relativistic kinetic energy. For (b), we will calculate the classical kinetic energy (which would be close to the relativistic value if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> were less than a few percent of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>) and see that it is not the same.<\/p>\n<p id=\"import-auto-id2837815\"><strong>Solution for (a) <\/strong><\/p>\n<ol id=\"fs-id1574907\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2546385\">Identify the knowns. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0654f4aad251855943737c7e0d273cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#48;&#125;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"81\" style=\"vertical-align: 0px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43371ec0dd474c56f0f40123f63ffb34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -3px;\" \/><\/li>\n<li id=\"import-auto-id2597542\">Identify the unknown. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-55813247d68384087a68f48a446e885b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/><\/li>\n<li id=\"import-auto-id3081018\">Choose the appropriate equation. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7089cb27a98531de9f707da3ae7b4a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/><\/li>\n<li id=\"import-auto-id2595330\">Plug the knowns into the equation.\n<p id=\"import-auto-id1489564\">First calculate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfd339d0f13026ff7af56aa6f129380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/>. We will carry extra digits because this is an intermediate calculation.<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cfb868227326916252a8e01c3ae5ccb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#48;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#56;&#56;&#56;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"143\" style=\"vertical-align: -56px;\" \/><\/div>\n<p id=\"import-auto-id1535841\">Next, we use this value to calculate the kinetic energy.<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-26d7f8844fca1e7b9d56259beae5cdbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#46;&#48;&#56;&#56;&#56;&#125;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#46;&#49;&#49;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#52;&#46;&#57;&#57;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"416\" style=\"vertical-align: -24px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id2003431\">Convert units.\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91b9d6b25d251fec455601d8add3f71e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#57;&#57;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#77;&#101;&#86;&#125;&#125;&#123;&#49;&#46;&#54;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#46;&#49;&#50;&#32;&#77;&#101;&#86;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"279\" style=\"vertical-align: -13px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p><strong>Solution for (b) <\/strong><\/p>\n<ol id=\"fs-id2584774\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2609096\">List the knowns. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0654f4aad251855943737c7e0d273cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#48;&#125;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"81\" style=\"vertical-align: 0px;\" \/>; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43371ec0dd474c56f0f40123f63ffb34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -3px;\" \/><\/li>\n<li id=\"import-auto-id2660805\">List the unknown. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-709e40e41f013af51396f99ac556037a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#108;&#97;&#115;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"53\" style=\"vertical-align: -4px;\" \/><\/li>\n<li id=\"import-auto-id1591594\">Choose the appropriate equation. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c2bd0805041519f9dbcb510378c511b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#108;&#97;&#115;&#115;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#118;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"120\" style=\"vertical-align: -6px;\" \/><\/li>\n<li id=\"import-auto-id3143593\">Plug the knowns into the equation.\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18d8771cd534821d7265a0a2256caa3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#108;&#97;&#115;&#115;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#118;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#46;&#57;&#57;&#48;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#52;&#46;&#48;&#50;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"411\" style=\"vertical-align: -24px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1505723\">Convert units.\n<div data-type=\"equation\" class=\"equation\" id=\"eip-719\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1a916b704c1bb5f5b4294fba823fc9cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#108;&#97;&#115;&#115;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#52;&#46;&#48;&#50;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#77;&#101;&#86;&#125;&#125;&#123;&#49;&#46;&#54;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#50;&#53;&#49;&#32;&#77;&#101;&#86;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"281\" style=\"vertical-align: -13px;\" \/><\/div>\n<\/li>\n<\/ol>\n<p id=\"import-auto-id2771680\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1523780\">As might be expected, since the velocity is 99.0% of the speed of light, the classical kinetic energy is significantly off from the correct relativistic value. Note also that the classical value is much smaller than the relativistic value. In fact, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-df19e41612af6b53c97eb27cad24e1f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#108;&#97;&#115;&#115;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"160\" style=\"vertical-align: -6px;\" \/> here. This is some indication of how difficult it is to get a mass moving close to the speed of light. Much more energy is required than predicted classically. Some people interpret this extra energy as going into increasing the mass of the system, but, as discussed in <a href=\"\/contents\/cb474108-fe67-4904-bdbe-441b271ec10f\">Relativistic Momentum<\/a>, this cannot be verified unambiguously. What is certain is that ever-increasing amounts of energy are needed to get the velocity of a mass a little closer to that of light. An energy of 3 MeV is a very small amount for an electron, and it can be achieved with present-day particle accelerators. SLAC, for example, can accelerate electrons to over <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b1cf7360d688b4a069ade26c9ebe735d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#44;&#48;&#48;&#48;&#32;&#77;&#101;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"185\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<p id=\"import-auto-id2910636\">Is there any point in getting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> a little closer to c than 99.0% or 99.9%? The answer is yes. We learn a great deal by doing this. The energy that goes into a high-velocity mass can be converted to any other form, including into entirely new masses. (See <a href=\"#import-auto-id3098026\" class=\"autogenerated-content\">(Figure)<\/a>.) Most of what we know about the substructure of matter and the collection of exotic short-lived particles in nature has been learned this way. Particles are accelerated to extremely relativistic energies and made to collide with other particles, producing totally new species of particles. Patterns in the characteristics of these previously unknown particles hint at a basic substructure for all matter. These particles and some of their characteristics will be covered in <a href=\"\/contents\/43a84abb-7daf-467e-ae21-b1b74baab670\">Particle Physics<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3098026\">\n<div class=\"bc-figcaption figcaption\">The Fermi National Accelerator Laboratory, near Batavia, Illinois, was a subatomic particle collider that accelerated protons and antiprotons to attain energies up to 1 Tev (a trillion electronvolts). The circular ponds near the rings were built to dissipate waste heat. This accelerator was shut down in September 2011. (credit: Fermilab, Reidar Hahn)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3098027\" data-alt=\"An aerial view of the Fermi National Accelerator Laboratory. The accelerator has two large, ring shaped structures. There are circular ponds near the rings.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_29_06_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"An aerial view of the Fermi National Accelerator Laboratory. The accelerator has two large, ring shaped structures. There are circular ponds near the rings.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2573402\">\n<h1 data-type=\"title\">Relativistic Energy and Momentum<\/h1>\n<p id=\"import-auto-id2724159\">We know classically that kinetic energy and momentum are related to each other, since<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3e70296cfeef8c19966a18c4aa7574c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#108;&#97;&#115;&#115;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#109;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#118;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#109;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#118;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"234\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2603540\">Relativistically, we can obtain a relationship between energy and momentum by algebraically manipulating their definitions. This produces<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-035a206232c7f5f56e8c48e4563e5e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#99;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"168\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2634770\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the relativistic total energy and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> is the relativistic momentum. This relationship between relativistic energy and relativistic momentum is more complicated than the classical, but we can gain some interesting new insights by examining it. First, total energy is related to momentum and rest mass. At rest, momentum is zero, and the equation gives the total energy to be the rest energy <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2ca930fd3e1c1a242725e5324be20573_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"30\" style=\"vertical-align: 0px;\" \/> (so this equation is consistent with the discussion of rest energy above). However, as the mass is accelerated, its momentum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> increases, thus increasing the total energy. At sufficiently high velocities, the rest energy term <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0407e8d13e7ea7dc55fed7c9da8f4de0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"52\" style=\"vertical-align: -7px;\" \/> becomes negligible compared with the momentum term <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f4f9df37843c192c2540f1e9b15499d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#112;&#99;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"38\" style=\"vertical-align: -4px;\" \/>; thus, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-913887410c81165e24e917a9f0a6e4aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#112;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -3px;\" \/> at extremely relativistic velocities.<\/p>\n<p id=\"import-auto-id1357067\">If we consider momentum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> to be distinct from mass, we can determine the implications of the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-035a206232c7f5f56e8c48e4563e5e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#99;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"168\" style=\"vertical-align: -7px;\" \/>  for a particle that has no mass. If we take <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/> to be zero in this equation, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-740e26ba612663d0b436561ac8c13c26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -3px;\" \/>, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-efc4a78d0ea66b45e257c226eaed7a3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#69;&#47;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -5px;\" \/>. Massless particles have this momentum. There are several massless particles found in nature, including photons (these are quanta of electromagnetic radiation). Another implication is that a massless particle must travel at speed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> and only at speed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>. While it is beyond the scope of this text to examine the relationship in the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-035a206232c7f5f56e8c48e4563e5e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#99;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"168\" style=\"vertical-align: -7px;\" \/> in detail, we can see that the relationship has important implications in special relativity.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2607341\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Problem-Solving Strategies for Relativity<\/div>\n<ol id=\"fs-id2677927\" data-number-style=\"arabic\" class=\"stepwise\">\n<li id=\"import-auto-id3067421\"><em data-effect=\"italics\"> Examine the situation to determine that it is necessary to use relativity<\/em>. Relativistic effects are related to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7542a73db72a1290e0d44df531398e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"85\" style=\"vertical-align: -21px;\" \/>, the quantitative relativistic factor. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfd339d0f13026ff7af56aa6f129380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> is very close to 1, then relativistic effects are small and differ very little from the usually easier classical calculations.<\/li>\n<li id=\"import-auto-id1675588\"><em data-effect=\"italics\"> Identify exactly what needs to be determined in the problem (identify the unknowns).<\/em><\/li>\n<li id=\"import-auto-id1364684\"><em data-effect=\"italics\"> Make a list of what is given or can be inferred from the problem as stated (identify the knowns).<\/em> Look in particular for information on relative velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/li>\n<li id=\"import-auto-id2608837\"><em data-effect=\"italics\"> Make certain you understand the conceptual aspects of the problem before making any calculations.<\/em> Decide, for example, which observer sees time dilated or length contracted before plugging into equations. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.<\/li>\n<li id=\"import-auto-id2515324\"><em data-effect=\"italics\"> Determine the primary type of calculation to be done to find the unknowns identified above.<\/em> You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.<\/li>\n<li id=\"import-auto-id2651965\"><em data-effect=\"italics\"> Do not round off during the calculation.<\/em> As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem, but do not use a rounded number in a subsequent calculation.<\/li>\n<li id=\"import-auto-id1357009\"><em data-effect=\"italics\"> Check the answer to see if it is reasonable: Does it make sense?<\/em> This may be more difficult for relativity, since we do not encounter it directly. But you can look for velocities greater than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"eip-id1537860\">\n<p id=\"eip-id1537861\">A photon decays into an electron-positron pair. What is the kinetic energy of the electron if its speed is  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4264cc2931e432766c3925931176facd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#57;&#57;&#50;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\" \/>?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id1537883\" data-label=\"\">\n<div data-type=\"title\">Answer<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1537888\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7893ab84160b1dde54e1ca65a9b242e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#57;&#57;&#50;&#125;&#99;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#46;&#49;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#54;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"119\" width=\"584\" style=\"vertical-align: -55px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3107783\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2999106\">\n<li id=\"import-auto-id2747191\">Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.<\/li>\n<li id=\"import-auto-id2944323\">Total Energy is defined as:<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d2a4cd7bb455360758a0664cbf9c012_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/>, where <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7542a73db72a1290e0d44df531398e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"85\" style=\"vertical-align: -21px;\" \/>.<\/p>\n<\/li>\n<li id=\"import-auto-id2821670\">Rest energy is<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-51e9548d34219b8907c811a780cdd4f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -3px;\" \/>,  meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.<\/li>\n<li id=\"import-auto-id2972722\">We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy.<\/li>\n<li id=\"import-auto-id3143536\">The relativistic work-energy theorem is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab2fd45bdaae462293acaabb254b8a08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#87;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#69;&#45;&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"344\" style=\"vertical-align: -4px;\" \/>.<\/li>\n<li>Relativistically,<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1edf594fea68219e837a741ffb90580c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#87;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"101\" style=\"vertical-align: -4px;\" \/> ,<br \/>\nwhere  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-55813247d68384087a68f48a446e885b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/> is the relativistic kinetic energy.<\/li>\n<li id=\"import-auto-id2768271\">Relativistic kinetic energy is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7089cb27a98531de9f707da3ae7b4a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7542a73db72a1290e0d44df531398e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"85\" style=\"vertical-align: -21px;\" \/>. At low velocities, relativistic kinetic energy reduces to classical kinetic energy.<\/li>\n<li id=\"import-auto-id3103866\"><strong>No object with mass can attain the speed of light<\/strong> because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.<\/li>\n<li id=\"import-auto-id2072642\">The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-55ea5723607e3b49f069e2a3ff911398_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#112;&#99;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"160\" style=\"vertical-align: -7px;\" \/> relates the relativistic total energy\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> and the relativistic momentum<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/>. At extremely high velocities, the rest energy<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2ca930fd3e1c1a242725e5324be20573_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"30\" style=\"vertical-align: 0px;\" \/> becomes negligible, and<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-913887410c81165e24e917a9f0a6e4aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#112;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id3113705\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1563904\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1546041\">\n<p id=\"import-auto-id2762569\">How are the classical laws of conservation of energy and conservation of mass modified by modern relativity?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2962135\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2962136\">\n<p id=\"import-auto-id2712204\">What happens to the mass of water in a pot when it cools, assuming no molecules escape or are added? Is this observable in practice? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2985598\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2527016\">\n<p id=\"import-auto-id1256380\">Consider a thought experiment. You place an expanded balloon of air on weighing scales outside in the early morning. The balloon stays on the scales and you are able to measure changes in its mass. Does the mass of the balloon change as the day progresses? Discuss the difficulties in carrying out this experiment.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2768272\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3459817\">\n<p id=\"import-auto-id1550823\">The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2634587\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2634588\">\n<p id=\"import-auto-id2634720\">We know that the velocity of an object with mass has an upper limit of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>. Is there an upper limit on its momentum? Its energy? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2834533\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2834534\">\n<p id=\"import-auto-id2602233\">Given the fact that light travels at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, can it have mass? Explain.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2633453\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2633454\">\n<p id=\"import-auto-id2963402\">If you use an Earth-based telescope to project a laser beam onto the Moon, you can move the spot across the Moon\u2019s surface at a velocity greater than the speed of light. Does this violate modern relativity? (Note that light is being sent from the Earth to the Moon, not across the surface of the Moon.)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id3164432\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3084928\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2944361\">\n<p id=\"import-auto-id1999778\">What is the rest energy of an electron, given its mass is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7d6fa3fd511a85e751573877fa403984_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -3px;\" \/>? Give your answer in joules and MeV.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1959372\">\n<p id=\"import-auto-id1433005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d8dd0007a97efc776e46958e4ad72a35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#46;&#50;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2724228\">0.512 MeV<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2721624\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2721625\">\n<p id=\"import-auto-id2573635\">Find the rest energy in joules and MeV of a proton, given its mass is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0fc3295fe81c9d141a64ad80eeaf7b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2681831\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2573985\">\n<p id=\"import-auto-id2644941\">If the rest energies of a proton and a neutron (the two constituents of nuclei) are 938.3 and 939.6 MeV respectively, what is the difference in their masses in kilograms?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1985675\">\n<p id=\"import-auto-id2675917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b52b929763da87c1eb847fd49a89399_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2720780\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2720781\">\n<p id=\"import-auto-id2736047\">The Big Bang that began the universe is estimated to have released <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-76246d33227f3ef8b811f6333317d511_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -1px;\" \/> of energy. How many stars could half this energy create, assuming the average star\u2019s mass is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c4554a5f5d47a17dcc69ff78a95905d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -3px;\" \/>?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2512400\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1362793\">\n<p id=\"import-auto-id3138939\">A supernova explosion of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793f2f2743b167b81e082ab3cdca6ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -3px;\" \/> star produces <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a4a038a59d24d0df3ced8eec3af803bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -1px;\" \/> of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-62b2ccaea28bc21890663c05d10d3d1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#47;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -5px;\" \/> of mass destroyed to the original mass of the star?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1605062\">\n<p id=\"import-auto-id1565184\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f7c22a716c09a78b280f72a2257db448_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"85\" style=\"vertical-align: -3px;\" \/>\n    <\/p>\n<p id=\"import-auto-id1563498\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39e0fc6fae8d2604b6fb69290aeae4f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -1px;\" \/>\n    <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2572932\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2572933\">\n<p id=\"import-auto-id2912658\">(a) Using data from <a href=\"\/contents\/c5adb012-fbab-41a7-b5b2-4644984097fc#import-auto-id2866785\" class=\"autogenerated-content\">(Figure)<\/a>, calculate the mass converted to energy by the fission of 1.00 kg of uranium. (b) What is the ratio of mass destroyed to the original mass, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-62b2ccaea28bc21890663c05d10d3d1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#47;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -5px;\" \/>?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1539307\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1539308\">\n<p id=\"import-auto-id1363138\">(a) Using data from <a href=\"\/contents\/c5adb012-fbab-41a7-b5b2-4644984097fc#import-auto-id2866785\" class=\"autogenerated-content\">(Figure)<\/a>, calculate the amount of mass converted to energy by the fusion of 1.00 kg of hydrogen. (b) What is the ratio of mass destroyed to the original mass, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-62b2ccaea28bc21890663c05d10d3d1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#47;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -5px;\" \/>? (c) How does this compare with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-62b2ccaea28bc21890663c05d10d3d1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#47;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -5px;\" \/> for the fission of 1.00 kg of uranium?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id3084938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7d0b9f782dca6567caafc2f3c87fb439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -3px;\" \/><\/p>\n<p id=\"import-auto-id2631704\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c20dc1fc9308bace3756b8b62892478d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#49;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id3143597\">The ratio is greater for hydrogen.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1780374\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1780375\">\n<p id=\"import-auto-id1612975\">There is approximately <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0b9f53dbf39cb586cab54070842c59e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -1px;\" \/> of energy available from fusion of hydrogen in the world\u2019s oceans. (a) If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dcf6e271334315b2689ef722670270f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -1px;\" \/> of this energy were utilized, what would be the decrease in mass of the oceans? (b) How great a volume of water does this correspond to? (c) Comment on whether this is a significant fraction of the total mass of the oceans.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1350397\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1350398\">\n<p id=\"import-auto-id2575969\">A muon has a rest mass energy of 105.7 MeV, and it decays into an electron and a massless particle. (a) If all the lost mass is converted into the electron\u2019s kinetic energy, find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfd339d0f13026ff7af56aa6f129380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> for the electron. (b) What is the electron\u2019s velocity?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2560231\">\n<p id=\"import-auto-id2003279\">208<\/p>\n<p id=\"import-auto-id1956060\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae4f2ed376f5f0086dead860f7801082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#57;&#57;&#57;&#57;&#56;&#56;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"76\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2702554\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2702555\">\n<p id=\"import-auto-id2607634\">A <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3868841096278f98ab5b439f54df304a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>-meson is a particle that decays into a muon and a massless particle. The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3868841096278f98ab5b439f54df304a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>-meson has a rest mass energy of 139.6 MeV, and the muon has a rest mass energy of 105.7 MeV. Suppose the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3868841096278f98ab5b439f54df304a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>-meson is at rest and all of the missing mass goes into the muon\u2019s kinetic energy. How fast will the muon move?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3358455\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3358456\">\n<p id=\"import-auto-id2651913\">(a) Calculate the relativistic kinetic energy of a 1000-kg car moving at 30.0 m\/s if the speed of light were only 45.0 m\/s. (b) Find the ratio of the relativistic kinetic energy to classical.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2773029\">\n<p id=\"import-auto-id1440827\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f1a7b81c16676a04f9b9c8fa88af046e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#57;&#50;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id1573461\">1.54<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2753673\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2753674\">\n<p id=\"import-auto-id2594708\">Alpha decay is nuclear decay in which a helium nucleus is emitted. If the helium nucleus has a mass of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-529326f1d77c4202397ec8868d81d074_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -3px;\" \/> and is given 5.00 MeV of kinetic energy, what is its velocity?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3195968\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3195969\">\n<p id=\"import-auto-id3204662\">(a) Beta decay is nuclear decay in which an electron is emitted. If the electron is given 0.750 MeV of kinetic energy, what is its velocity? (b) Comment on how the high velocity is consistent with the kinetic energy as it compares to the rest mass energy of the electron.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3137588\">\n<p>(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5289c2adb816dc6a6615cc93d8acf074_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#49;&#52;&#125;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/>\n    <\/p>\n<p id=\"import-auto-id2783066\">(b) The<br \/>\nrest<br \/>\nmass<br \/>\nenergy<br \/>\nof<br \/>\nan<br \/>\nelectron<br \/>\nis<br \/>\n0.511<br \/>\nMeV,<br \/>\nso<br \/>\nthe<br \/>\nkinetic<br \/>\nenergy<br \/>\nis<br \/>\napproximately<br \/>\n150%<br \/>\nof<br \/>\nthe<br \/>\nrest<br \/>\nmass<br \/>\nenergy.<br \/>\nThe<br \/>\nelectron<br \/>\nshould<br \/>\nbe<br \/>\ntraveling<br \/>\nclose<br \/>\nto<br \/>\nthe<br \/>\nspeed<br \/>\nof<br \/>\nlight.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2600562\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2600563\">\n<p id=\"import-auto-id2592460\">A positron is an antimatter version of the electron, having exactly the same mass. When a positron and an electron meet, they annihilate, converting all of their mass into energy. (a) Find the energy released, assuming negligible kinetic energy before the annihilation. (b) If this energy is given to a proton in the form of kinetic energy, what is its velocity? (c) If this energy is given to another electron in the form of kinetic energy, what is its velocity?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1284916\">\n<p id=\"import-auto-id2595222\">What is the kinetic energy in MeV of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3868841096278f98ab5b439f54df304a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>-meson that lives <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fab93d02b56436f9ab89bf321bcb737e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#125;&times;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/> as measured in the laboratory, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4fb2924a2c2589e18d7883a3e4f93846_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#52;&#48;&#125;&times;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -1px;\" \/> when at rest relative to an observer, given that its rest energy is 135 MeV?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2723222\">\n<p id=\"import-auto-id2724319\">90.0 MeV<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2725006\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2725008\">\n<p id=\"import-auto-id3101739\">Find the kinetic energy in MeV of a neutron with a measured life span of 2065 s, given its rest energy is 939.6 MeV, and rest life span is 900s.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2500966\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2500967\">\n<p id=\"import-auto-id1656559\">(a) Show that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f64a7d1a22ae32132ca75c70db484c26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#112;&#99;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#47;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"179\" style=\"vertical-align: -7px;\" \/>. This means that at large velocities <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-93d82fe19f117c21ed659f9eed83cf92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#112;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#62;&#62;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -3px;\" \/>. (b) Is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b868e37896b2a81e42087d8920153e25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -3px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42c5f0854ebf3ea85760e735b3e1f868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" style=\"vertical-align: -4px;\" \/>, as for the astronaut discussed in the twin paradox?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3164547\">\n<p id=\"eip-id1432763\">(a)     <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f2c7c3c671fadcff4d8ee5f7f3fdfbc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#125;&#123;&#69;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#123;&#99;&#125;&#94;&#123;&#52;&#125;&#61;&#123;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#125;&#94;&#123;&#50;&#125;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#123;&#99;&#125;&#94;&#123;&#52;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#32;&#116;&#104;&#97;&#116;&#125;&#92;&#92;&#32;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#123;&#99;&#125;&#94;&#123;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#32;&#97;&#110;&#100;&#32;&#116;&#104;&#101;&#114;&#101;&#102;&#111;&#114;&#101;&#125;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#99;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#109;&#97;&#116;&#104;&#105;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#99;&#125;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#123;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"77\" width=\"285\" style=\"vertical-align: -34px;\" \/><\/p>\n<p id=\"import-auto-id3087534\">(b) yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2834155\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2608664\">\n<p id=\"import-auto-id2509029\">One cosmic ray neutron has a velocity of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7bc65e814b34ac66a5736b3465058457_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#50;&#53;&#48;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/> relative to the Earth. (a) What is the neutron\u2019s total energy in MeV? (b) Find its momentum. (c) Is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b868e37896b2a81e42087d8920153e25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -3px;\" \/> in this situation? Discuss in terms of the equation given in part (a) of the previous problem.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2629153\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2629154\">\n<p id=\"import-auto-id3088688\">What is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfd339d0f13026ff7af56aa6f129380_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> for a proton having a mass energy of 938.3 MeV accelerated through an effective potential of 1.0 TV (teravolt) at Fermilab outside Chicago?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2942320\">\n<p id=\"import-auto-id2009307\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8d0793e9d951b782e0a8e0a2b948ed5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3164443\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3164444\">\n<p id=\"import-auto-id2676284\">(a) What is the effective accelerating potential for electrons at the Stanford Linear Accelerator, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-108884fecef3daf2ce32b4647d6ff0c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -4px;\" \/> for them? (b) What is their total energy (nearly the same as kinetic in this case) in GeV?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2911245\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2911246\">\n<p id=\"import-auto-id2502156\">(a) Using data from <a href=\"\/contents\/c5adb012-fbab-41a7-b5b2-4644984097fc#import-auto-id2866785\" class=\"autogenerated-content\">(Figure)<\/a>, find the mass destroyed when the energy in a barrel of crude oil is released. (b) Given these barrels contain 200 liters and assuming the density of crude oil is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cde7bbbe4d77000131d65989ffa5d295_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#53;&#48;&#32;&#107;&#103;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"76\" style=\"vertical-align: -4px;\" \/>, what is the ratio of mass destroyed to original mass, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-62b2ccaea28bc21890663c05d10d3d1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#109;&#47;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"55\" style=\"vertical-align: -5px;\" \/>?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3165931\">\n<p id=\"import-auto-id2923850\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e13097919d664aeda21fcff5f2ab4d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -3px;\" \/><\/p>\n<p id=\"import-auto-id2977369\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-df0fe33714ffb4f3036b85be86e5768c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#51;&#55;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3259500\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3259501\">\n<p id=\"import-auto-id2746932\">(a) Calculate the energy released by the destruction of 1.00 kg of mass. (b) How many kilograms could be lifted to a 10.0 km height by this amount of energy?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2656682\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2656683\">\n<p id=\"import-auto-id1415466\">A Van de Graaff accelerator utilizes a 50.0 MV potential difference to accelerate charged particles such as protons. (a) What is the velocity of a proton accelerated by such a potential? (b) An electron?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3040526\">\n<p id=\"import-auto-id1415472\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c06779238fb3ab2769e940fccdc49004_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#51;&#49;&#52;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"eip-id1461415\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fb46cb31fcdd2f92f2942ffc62e8895_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#46;&#57;&#57;&#57;&#57;&#53;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3143371\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3143372\">\n<p id=\"import-auto-id3371246\">Suppose you use an average of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ed74cd5c262eb4b99fa4d0b22d3aa65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#48;&#32;&#107;&#87;&middot;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: 0px;\" \/>  of electric energy per month in your home. (a) How long would 1.00 g of mass converted to electric energy with an efficiency of 38.0% last you? (b) How many homes could be supplied at the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ed74cd5c262eb4b99fa4d0b22d3aa65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#48;&#32;&#107;&#87;&middot;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: 0px;\" \/> per month rate for one year by the energy from the described mass conversion?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1946210\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3158295\">\n<p id=\"import-auto-id1356748\">(a) A nuclear power plant converts energy from nuclear fission into electricity with an efficiency of 35.0%. How much mass is destroyed in one year to produce a continuous 1000 MW of electric power? (b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-294f1add9a7460a12e7e595cb64de658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -3px;\" \/>?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2836488\">\n<p id=\"import-auto-id2657369\">(a) 1.00 kg<\/p>\n<p id=\"eip-id2744805\">(b) This much mass would be measurable, but probably not observable just by looking because it is 0.01% of the total mass.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3159094\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3159095\">\n<p>Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket\u2019s mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b07528e1202ce444f8444d22d083e89e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#48;&#48;&times;&#123;&#49;&#48;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -3px;\" \/> (100 tons), what total yield nuclear explosion in tons of TNT is needed?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2693263\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2693264\">\n<p id=\"import-auto-id3152001\">The Sun produces energy at a rate of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66e36f11348ed6506f5c09579999934e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/> W by the fusion of hydrogen. (a) How many kilograms of hydrogen undergo fusion each second? (b) If the Sun is 90.0% hydrogen and half of this can undergo fusion before the Sun changes character, how long could it produce energy at its current rate? (c) How many kilograms of mass is the Sun losing per second? (d) What fraction of its mass will it have lost in the time found in part (b)?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2602531\">\n<p id=\"import-auto-id3084632\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-62cfc8e6d9f9e57cdcffb3b6d9cb5cc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -4px;\" \/>\n    <\/p>\n<p id=\"import-auto-id3087172\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2b97316453599e3112466dd196113a41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -3px;\" \/>\n    <\/p>\n<p id=\"import-auto-id2761625\">(c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c7b8c8eb3e7e26b51c5a72af466048e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -3px;\" \/>\n    <\/p>\n<p id=\"import-auto-id2693455\">(d) 0.32%<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2791387\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2791388\">\n<p id=\"eip-id3377332\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id3109827\">A proton has a mass of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0fc3295fe81c9d141a64ad80eeaf7b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"96\" style=\"vertical-align: -3px;\" \/>. A physicist measures the proton\u2019s total energy to be 50.0 MeV. (a) What is the proton\u2019s kinetic energy? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1936682\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1936683\">\n<p id=\"import-auto-id3067560\"><strong>Construct Your Own Problem<\/strong><\/p>\n<p id=\"eip-id2312930\">Consider a highly relativistic particle. Discuss what is meant by the term \u201chighly relativistic.\u201d (Note that, in part, it means that the particle cannot be massless.) Construct a problem in which you calculate the wavelength of such a particle and show that it is very nearly the same as the wavelength of a massless particle, such as a photon, with the same energy. Among the things to be considered are the rest energy of the particle (it should be a known particle) and its total energy, which should be large compared to its rest energy.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2929431\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2929432\">\n<p id=\"import-auto-id2657479\"><strong>Construct Your Own Problem<\/strong><\/p>\n<p id=\"eip-id1632086\">Consider an astronaut traveling to another star at a relativistic velocity. Construct a problem in which you calculate the time for the trip as observed on the Earth and as observed by the astronaut. Also calculate the amount of mass that must be converted to energy to get the astronaut and ship to the velocity travelled. Among the things to be considered are the distance to the star, the velocity, and the mass of the astronaut and ship. Unless your instructor directs you otherwise, do not include any energy given to other masses, such as rocket propellants.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1364291\">\n<dt>total energy<\/dt>\n<dd id=\"fs-id1360334\"> defined as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d2a4cd7bb455360758a0664cbf9c012_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -4px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7542a73db72a1290e0d44df531398e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"85\" style=\"vertical-align: -21px;\" \/><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1781799\">\n<dt>rest energy<\/dt>\n<dd id=\"fs-id2677386\">the energy stored in an object at rest: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-51e9548d34219b8907c811a780cdd4f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -3px;\" \/><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3167814\">\n<dt>relativistic kinetic energy<\/dt>\n<dd id=\"fs-id1359660\">the kinetic energy of an object moving at relativistic speeds: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7089cb27a98531de9f707da3ae7b4a0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#69;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#108;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#109;&#99;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7542a73db72a1290e0d44df531398e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#103;&#97;&#109;&#109;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"85\" style=\"vertical-align: -21px;\" \/><\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1559","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1523,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1559","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1559\/revisions"}],"predecessor-version":[{"id":1560,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1559\/revisions\/1560"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1523"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1559\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1559"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1559"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1559"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}