{"id":421,"date":"2017-10-27T16:29:49","date_gmt":"2017-10-27T16:29:49","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/linear-momentum-and-force\/"},"modified":"2017-11-08T03:24:34","modified_gmt":"2017-11-08T03:24:34","slug":"linear-momentum-and-force","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/linear-momentum-and-force\/","title":{"raw":"Linear Momentum and Force","rendered":"Linear Momentum and Force"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Define linear momentum.<\/li>\n<li>Explain the relationship between momentum and force.<\/li>\n<li>State Newton\u2019s second law of motion in terms of momentum.<\/li>\n<li>Calculate momentum given mass and velocity.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3586260\">\n<h1 data-type=\"title\">Linear Momentum<\/h1>\n<p id=\"import-auto-id1303032\">The scientific definition of linear momentum is consistent with most people\u2019s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. <span data-type=\"term\" id=\"import-auto-id1343909\">Linear momentum<\/span> is defined as the product of a system\u2019s mass multiplied by its velocity. In symbols, linear momentum is expressed as<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-927\">[latex]\\mathbf{p}=m\\mathbf{\\text{v}}.[\/latex]<\/div>\n<p id=\"import-auto-id1255821\">Momentum is directly proportional to the object\u2019s mass and also its velocity. Thus the greater an object\u2019s mass or the greater its velocity, the greater its momentum. Momentum [latex]\\mathbf{p}[\/latex] is a vector having the same direction as the velocity [latex]\\mathbf{\\text{v}}[\/latex]. The SI unit for momentum is [latex]\\text{kg}\u00b7\\text{m\/s}[\/latex]. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1134069\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Linear Momentum<\/div>\n<p id=\"import-auto-id1120695\">Linear momentum is defined as the product of a system\u2019s mass multiplied by its velocity:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-695\">[latex]\\mathbf{p}=m\\mathbf{\\text{v}}.[\/latex]<\/div>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1356444\">\n<div data-type=\"title\" class=\"title\">Calculating Momentum: A Football Player and a Football<\/div>\n<p id=\"import-auto-id1171239\">(a) Calculate the momentum of a 110-kg football player running at 8.00 m\/s. (b) Compare the player\u2019s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m\/s.<\/p>\n<p id=\"import-auto-id1250149\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1162402\">No information is given regarding direction, and so we can calculate only the magnitude of the momentum, [latex]p[\/latex]. (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]p=\\text{mv}[\/latex]<\/div>\n<p id=\"import-auto-id1109788\">when only magnitudes are considered.<\/p>\n<p id=\"import-auto-id1445232\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1143096\">To determine the momentum of the player, substitute the known values for the player\u2019s mass and speed into the equation.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-926\">[latex]{p}_{\\text{player}}=\\left(\\text{110 kg}\\right)\\left(8\\text{.}\\text{00 m\/s}\\right)=\\text{880 kg}\u00b7\\text{m\/s}[\/latex]<\/div>\n<p id=\"eip-834\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1250861\">To determine the momentum of the ball, substitute the known values for the ball\u2019s mass and speed into the equation.<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{p}_{\\text{ball}}=\\left(\\text{0.410 kg}\\right)\\left(\\text{25.0 m\/s}\\right)=\\text{10.3 kg}\u00b7\\text{m\/s}[\/latex]<\/div>\n<p id=\"import-auto-id1524278\">The ratio of the player\u2019s momentum to that of the ball is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-45\">[latex]\\frac{{p}_{\\text{player}}}{{p}_{\\text{ball}}}=\\frac{\\text{880}}{\\text{10}\\text{.}3}=\\text{85}\\text{.}9.[\/latex]<\/div>\n<p id=\"import-auto-id1187619\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1251506\">Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player\u2019s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1342888\">\n<h1 data-type=\"title\">Momentum and Newton\u2019s Second Law<\/h1>\n<p id=\"import-auto-id1498551\">The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the \u201cquantity of motion.\u201d Newton actually stated his <span data-type=\"term\" id=\"import-auto-id1513532\">second law of motion<\/span> in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-96\">[latex]{\\mathbf{F}}_{\\text{net}}=\\frac{\\Delta \\mathbf{p}}{\\Delta t},[\/latex]<\/div>\n<p id=\"import-auto-id1498518\">where [latex]{\\mathbf{F}}_{\\text{net}}[\/latex] is the net external force, [latex]\\Delta \\mathbf{p}[\/latex] is the change in momentum, and [latex]\\Delta t[\/latex] is the change in time.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1440030\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Newton\u2019s Second Law of Motion in Terms of Momentum<\/div>\n<p id=\"import-auto-id1498318\">The net external force equals the change in momentum of a system divided by the time over which it changes.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-763\">[latex]{\\mathbf{F}}_{\\text{net}}=\\frac{\\Delta \\mathbf{p}}{\\Delta t}[\/latex]<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1535269\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Force and Momentum<\/div>\n<p id=\"import-auto-id1348915\">Force and momentum are intimately related. Force acting over time can change momentum, and Newton\u2019s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics.<\/p>\n<\/div>\n<p id=\"import-auto-id1404910\">This statement of Newton\u2019s second law of motion includes the more familiar [latex]{\\mathbf{F}}_{\\text{net}}\\text{=}m\\mathbf{a}[\/latex] as a special case. We can derive this form as follows. First, note that the change in momentum [latex]\\Delta \\mathbf{p}[\/latex] is given by <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-210\">[latex]\\Delta \\mathbf{p}=\\Delta \\left(m\\mathbf{v}\\right).[\/latex]<\/div>\n<p id=\"import-auto-id1235381\">If the mass of the system is constant, then<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-316\">[latex]\\Delta \\left(m\\mathbf{v}\\right)=m\\Delta \\mathbf{v}.[\/latex]<\/div>\n<p id=\"import-auto-id1269263\">So that for constant mass, Newton\u2019s second law of motion becomes<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-844\">[latex]{\\mathbf{F}}_{\\text{net}}=\\frac{\\Delta \\mathbf{p}}{\\Delta t}=\\frac{m\\Delta \\mathbf{v}}{\\Delta t}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1357595\">Because [latex]\\frac{\\Delta \\mathbf{v}}{\\Delta t}=\\mathbf{a}[\/latex], we get the familiar equation<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-17\">[latex]{\\mathbf{F}}_{\\text{net}}\\text{=}m\\mathbf{a}[\/latex]<\/div>\n<p id=\"import-auto-id1100094\"><em data-effect=\"italics\">when the mass of the system is constant<\/em>.<\/p>\n<p id=\"import-auto-id1305125\">Newton\u2019s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail<strong data-effect=\"bold\">;<\/strong> however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1523425\">\n<div data-type=\"title\" class=\"title\">Calculating Force: Venus Williams\u2019 Racquet<\/div>\n<p id=\"import-auto-id1372893\">During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women\u2019s match, reaching a speed of 58 m\/s (209 km\/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams\u2019 racquet, assuming that the ball\u2019s speed just after impact is 58 m\/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)? <\/p>\n<p id=\"import-auto-id1441326\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1484278\">This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton\u2019s second law stated in terms of momentum is then written as<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-344\">[latex]{\\mathbf{F}}_{\\text{net}}=\\frac{\\Delta \\mathbf{p}}{\\Delta t}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1108072\">As noted above, when mass is constant, the change in momentum is given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\Delta p=m\\Delta v=m\\left({v}_{f}-{v}_{i}\\right).[\/latex]<\/div>\n<p id=\"import-auto-id1173592\">In this example, the velocity just after impact and the change in time are given; thus, once [latex]\\Delta p[\/latex] is calculated, [latex]{F}_{\\text{net}}=\\frac{\\Delta p}{\\Delta t}[\/latex] can be used to find the force.<\/p>\n<p id=\"import-auto-id1285648\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1451495\">To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-107\">[latex]\\begin{array}{lll}\\Delta p&amp; =&amp; m\\left({v}_{f}\u2013{v}_{i}\\right)\\\\ &amp; =&amp; \\left(\\text{0.057 kg}\\right)\\left(\\text{58 m\/s}\u20130 m\/s\\right)\\\\ &amp; =&amp; 3\\text{.306 kg}\u00b7\\text{m\/s}\\approx \\text{3.3 kg}\u00b7\\text{m\/s}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1269051\">Now the magnitude of the net external force can determined by using [latex]{F}_{\\text{net}}=\\frac{\\Delta p}{\\Delta t}[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}{F}_{\\text{net}}&amp; =&amp; \\frac{\\Delta p}{\\Delta t}=\\frac{\\text{3.306 kg}\\cdot \\text{m\/s}}{5\\text{.}0\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}s}\\\\ &amp; =&amp; \\text{661 N}\\approx \\text{660 N,}\\end{array}[\/latex]<\/div>\n<p>where we have retained only two significant figures in the final step.<\/p>\n<p id=\"import-auto-id1488607\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1512339\">This quantity was the average force exerted by Venus Williams\u2019 racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using [latex]{F}_{\\text{net}}\\phantom{\\rule{0.15em}{0ex}}\\text{=}\\phantom{\\rule{0.15em}{0ex}}\\text{ma}[\/latex], but one additional step would be required compared with the strategy used in this example.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1521935\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1521940\">\n<li id=\"import-auto-id1499434\">Linear momentum (<em data-effect=\"italics\"><em data-effect=\"italics\">momentum<\/em><\/em> for brevity) is defined as the product of a system\u2019s mass multiplied by its velocity.<\/li>\n<li id=\"import-auto-id1489206\">In symbols, linear momentum [latex]\\mathbf{p}[\/latex] is defined to be\n<div data-type=\"equation\" class=\"equation\" id=\"eip-123\">[latex]\\mathbf{p}=m\\mathbf{v},[\/latex]<\/div>\n<p>    where <em data-effect=\"italics\">[latex]m[\/latex]<\/em> is the mass of the system and [latex]\\mathbf{v}[\/latex] is its velocity. <\/p><\/li>\n<li id=\"import-auto-id1512300\">The SI unit for momentum is [latex]\\text{kg}\u00b7\\text{m\/s}[\/latex].<\/li>\n<li id=\"import-auto-id1498973\">Newton\u2019s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes. <\/li>\n<li id=\"import-auto-id1498977\">In symbols, Newton\u2019s second law of motion is defined to be\n<div data-type=\"equation\" class=\"equation\" id=\"eip-370\">[latex]{\\mathbf{F}}_{\\text{net}}=\\frac{\\Delta \\mathbf{p}}{\\Delta t}\\text{,}[\/latex]<\/div>\n<p>[latex]{\\mathbf{F}}_{\\text{net}}[\/latex] is the net external force, [latex]\\Delta \\mathbf{p}[\/latex] is the change in momentum, and [latex]\\Delta t[\/latex] is the change time.<\/p><\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1522174\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522179\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1466205\">An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522190\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1522191\">\n<p id=\"import-auto-id1466209\">An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522201\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1522202\">\n<p id=\"import-auto-id1466212\"><strong>Professional Application<\/strong><\/p>\n<p id=\"eip-id974177\">Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522215\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1522216\">\n<p id=\"import-auto-id1489046\"> How can a small force impart the same momentum to an object as a large force?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1522226\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522230\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1522231\">\n<p id=\"import-auto-id1528696\">(a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of [latex]7\\text{.}\\text{50 m\/s}[\/latex]. (b) Compare the elephant\u2019s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of [latex]\\text{600 m\/s}[\/latex]. (c) What is the momentum of the 90.0-kg hunter running at [latex]7\\text{.}\\text{40 m\/s}[\/latex] after missing the elephant?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1493308\">\n<p id=\"import-auto-id1356784\">(a)<br>\n        [latex]\\text{1.50}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\cdot \\text{m\/s}[\/latex]<\/p>\n<p id=\"import-auto-id1489164\">(b) 625 to 1<\/p>\n<p id=\"import-auto-id1489166\">(c)<br>\n        [latex]6\\text{.}\\text{66}\u00d7{\\text{10}}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\\cdot \\text{m\/s}[\/latex]\n    <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1493437\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1493438\">\n<p id=\"import-auto-id1486822\">(a) What is the mass of a large ship that has a momentum of [latex]1\\text{.}\\text{60}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\u00b7\\text{m\/s}[\/latex], when the ship is moving at a speed of [latex]\\text{48.0 km\/h?}[\/latex] (b) Compare the ship\u2019s momentum to the momentum of a 1100-kg artillery shell fired at a speed of [latex]\\text{1200 m\/s}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1493544\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1493545\">\n<p id=\"import-auto-id1466181\">(a) At what speed would a [latex]2\\text{.}\\text{00}\u00d7{\\text{10}}^{4}\\text{-kg}[\/latex] airplane have to fly to have a momentum of [latex]1\\text{.}\\text{60}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\u00b7\\text{m\/s}[\/latex] (the same as the ship\u2019s momentum in the problem above)? (b) What is the plane\u2019s momentum when it is taking off at a speed of [latex]\\text{60.0 m\/s}[\/latex]? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1493671\">\n<p id=\"import-auto-id1512421\">(a) [latex]8\\text{.}\\text{00}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex]<\/p>\n<p id=\"import-auto-id1528742\">(b) [latex]1\\text{.}\\text{20}\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}\u00b7\\text{m\/s}[\/latex]<\/p>\n<p id=\"import-auto-id1466163\">(c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the ship, the ship will not recoil very much. The recoil would be [latex]-0\\text{.}\\text{0100 m\/s}[\/latex], which is probably not noticeable.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1493804\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1493805\">\n<p id=\"import-auto-id1489185\">(a) What is the momentum of a garbage truck that is [latex]1\\text{.}\\text{20}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex] and is moving at [latex]10\\text{.}\\text{0 m\/s}[\/latex]? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1493884\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1493885\">\n<p id=\"import-auto-id1488710\">A runaway train car that has a mass of 15,000 kg travels at a speed of [latex]5\\text{.4 m\/s}[\/latex] down a track. Compute the time required for a force of 1500 N to bring the car to rest.\n<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1542630\">\n<p id=\"import-auto-id1488755\">54 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1542640\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1542641\">\n<p id=\"import-auto-id1488764\">The mass of Earth is [latex]5\\text{.}\\text{972}\u00d7{10}^{\\text{24}}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex] and its orbital radius is an average of [latex]1\\text{.}\\text{496}\u00d7{10}^{\\text{11}}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]. Calculate its linear momentum.<\/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-id1499242\">\n<dt>linear momentum<\/dt>\n<dd id=\"fs-id1542740\">the product of mass and velocity<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1499250\">\n<dt>second law of motion<\/dt>\n<dd id=\"fs-id1542750\">physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Define linear momentum.<\/li>\n<li>Explain the relationship between momentum and force.<\/li>\n<li>State Newton\u2019s second law of motion in terms of momentum.<\/li>\n<li>Calculate momentum given mass and velocity.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3586260\">\n<h1 data-type=\"title\">Linear Momentum<\/h1>\n<p id=\"import-auto-id1303032\">The scientific definition of linear momentum is consistent with most people\u2019s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. <span data-type=\"term\" id=\"import-auto-id1343909\">Linear momentum<\/span> is defined as the product of a system\u2019s mass multiplied by its velocity. In symbols, linear momentum is expressed as<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-927\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f3d43a9e9697f59b45d8473dcf1697c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#61;&#109;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1255821\">Momentum is directly proportional to the object\u2019s mass and also its velocity. Thus the greater an object\u2019s mass or the greater its velocity, the greater its momentum. Momentum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e3c7a9253c3c0ae1fcdfdbd2d381f439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: -3px;\" \/> is a vector having the same direction as the velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2a0ddecb4ad20c4420849c699255b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"9\" style=\"vertical-align: 0px;\" \/>. The SI unit for momentum is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6016ed7e8a25b6fc8f6234c5d37318d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/>. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1134069\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Linear Momentum<\/div>\n<p id=\"import-auto-id1120695\">Linear momentum is defined as the product of a system\u2019s mass multiplied by its velocity:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-695\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f3d43a9e9697f59b45d8473dcf1697c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#61;&#109;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1356444\">\n<div data-type=\"title\" class=\"title\">Calculating Momentum: A Football Player and a Football<\/div>\n<p id=\"import-auto-id1171239\">(a) Calculate the momentum of a 110-kg football player running at 8.00 m\/s. (b) Compare the player\u2019s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m\/s.<\/p>\n<p id=\"import-auto-id1250149\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1162402\">No information is given regarding direction, and so we can calculate only the magnitude of the 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;\" \/>. (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes<\/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-d866fee32761bd97b77a29e36a608e36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1109788\">when only magnitudes are considered.<\/p>\n<p id=\"import-auto-id1445232\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1143096\">To determine the momentum of the player, substitute the known values for the player\u2019s mass and speed into the equation.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-926\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c8cf4ebac2272af93f0c70c1acfbf826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#108;&#97;&#121;&#101;&#114;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#48;&#32;&#107;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#56;&#48;&#32;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"325\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"eip-834\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1250861\">To determine the momentum of the ball, substitute the known values for the ball\u2019s mass and speed into the equation.<\/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-8b3851e8b0bbe5ff6c775cd4d5c89b8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#97;&#108;&#108;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#52;&#49;&#48;&#32;&#107;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#46;&#48;&#32;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#46;&#51;&#32;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"330\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1524278\">The ratio of the player\u2019s momentum to that of the ball is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-45\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bd703eaa1d6f206fab2931485cb0896b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#112;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#108;&#97;&#121;&#101;&#114;&#125;&#125;&#125;&#123;&#123;&#112;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#97;&#108;&#108;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#56;&#48;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"153\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1187619\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1251506\">Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player\u2019s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1342888\">\n<h1 data-type=\"title\">Momentum and Newton\u2019s Second Law<\/h1>\n<p id=\"import-auto-id1498551\">The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the \u201cquantity of motion.\u201d Newton actually stated his <span data-type=\"term\" id=\"import-auto-id1513532\">second law of motion<\/span> in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-96\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8f74dab675eafd921c33bd2b33672121_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1498518\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b52c16cde0b8a68dd12fa28a3a4db6ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"32\" style=\"vertical-align: -3px;\" \/> is the net external force, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7161e60e68eaf90be5138c4a3040070d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: -3px;\" \/> is the change in momentum, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0714636704a254c71bede042781bc57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> is the change in time.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1440030\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Newton\u2019s Second Law of Motion in Terms of Momentum<\/div>\n<p id=\"import-auto-id1498318\">The net external force equals the change in momentum of a system divided by the time over which it changes.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-763\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3cd6f697752fc989e075def3be119927_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1535269\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Force and Momentum<\/div>\n<p id=\"import-auto-id1348915\">Force and momentum are intimately related. Force acting over time can change momentum, and Newton\u2019s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics.<\/p>\n<\/div>\n<p id=\"import-auto-id1404910\">This statement of Newton\u2019s second law of motion includes the more familiar <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82e9bea4bb270c6c23c77219575f05c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#125;&#109;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"73\" style=\"vertical-align: -3px;\" \/> as a special case. We can derive this form as follows. First, note that the change in momentum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7161e60e68eaf90be5138c4a3040070d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: -3px;\" \/> is given by <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-210\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d6a210bd9aa509ca2060b48be8e0a7fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#61;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1235381\">If the mass of the system is constant, then<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-316\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3615f7c1f23a36d6a7de35c5a8798a3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#109;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1269263\">So that for constant mass, Newton\u2019s second law of motion becomes<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-844\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4ac578c33392e2511e3e79438663eb26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1357595\">Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-52a354b0f0ffa4c881ddb427996239c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/>, we get the familiar equation<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-17\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82e9bea4bb270c6c23c77219575f05c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#125;&#109;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"73\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1100094\"><em data-effect=\"italics\">when the mass of the system is constant<\/em>.<\/p>\n<p id=\"import-auto-id1305125\">Newton\u2019s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail<strong data-effect=\"bold\">;<\/strong> however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1523425\">\n<div data-type=\"title\" class=\"title\">Calculating Force: Venus Williams\u2019 Racquet<\/div>\n<p id=\"import-auto-id1372893\">During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women\u2019s match, reaching a speed of 58 m\/s (209 km\/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams\u2019 racquet, assuming that the ball\u2019s speed just after impact is 58 m\/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)? <\/p>\n<p id=\"import-auto-id1441326\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1484278\">This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton\u2019s second law stated in terms of momentum is then written as<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-344\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6f42d5b40ccee783de0e515072c9cd64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1108072\">As noted above, when mass is constant, the change in momentum is given by<\/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-3bdceb0089e5ea75eb5ae0bb00ac8f0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#112;&#61;&#109;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;&#61;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#102;&#125;&#45;&#123;&#118;&#125;&#95;&#123;&#105;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"203\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1173592\">In this example, the velocity just after impact and the change in time are given; thus, once <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-125a21e735bc4d008977a427433b1f51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -4px;\" \/> is calculated, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-deacc1a2d2aa1fdcc55e31a3da99c673_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#70;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#112;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"76\" style=\"vertical-align: -6px;\" \/> can be used to find the force.<\/p>\n<p id=\"import-auto-id1285648\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1451495\">To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-107\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b3cc9e71696ed3da8399b83e44bd4933_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;&#112;&#38;&#32;&#61;&#38;&#32;&#109;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#118;&#125;&#95;&#123;&#102;&#125;&#45;&#123;&#118;&#125;&#95;&#123;&#105;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#48;&#53;&#55;&#32;&#107;&#103;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#56;&#32;&#109;&#47;&#115;&#125;&#45;&#48;&#32;&#109;&#47;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#51;&#48;&#54;&#32;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#46;&#51;&#32;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"282\" style=\"vertical-align: -26px;\" \/><\/div>\n<p id=\"import-auto-id1269051\">Now the magnitude of the net external force can determined by using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-deacc1a2d2aa1fdcc55e31a3da99c673_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#70;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#112;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"76\" style=\"vertical-align: -6px;\" \/>:<\/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-00822bd0edf3d6d97ebd86bb2a83cf58_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;&#70;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#112;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#46;&#51;&#48;&#54;&#32;&#107;&#103;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#123;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#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;&#115;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#54;&#49;&#32;&#78;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#54;&#48;&#32;&#78;&#44;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"203\" style=\"vertical-align: -16px;\" \/><\/div>\n<p>where we have retained only two significant figures in the final step.<\/p>\n<p id=\"import-auto-id1488607\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1512339\">This quantity was the average force exerted by Venus Williams\u2019 racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b52d3deff82895bd3613f033175b5544_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#70;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"74\" style=\"vertical-align: -3px;\" \/>, but one additional step would be required compared with the strategy used in this example.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1521935\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1521940\">\n<li id=\"import-auto-id1499434\">Linear momentum (<em data-effect=\"italics\"><em data-effect=\"italics\">momentum<\/em><\/em> for brevity) is defined as the product of a system\u2019s mass multiplied by its velocity.<\/li>\n<li id=\"import-auto-id1489206\">In symbols, linear momentum <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e3c7a9253c3c0ae1fcdfdbd2d381f439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"11\" style=\"vertical-align: -3px;\" \/> is defined to be\n<div data-type=\"equation\" class=\"equation\" id=\"eip-123\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa2a313954ec06be0b13fbd4d2997394_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#61;&#109;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>    where <em data-effect=\"italics\"><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;\" \/><\/em> is the mass of the system and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1fb42cb2b0083b64704052f6366c336f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"11\" style=\"vertical-align: 0px;\" \/> is its velocity. <\/p>\n<\/li>\n<li id=\"import-auto-id1512300\">The SI unit for momentum is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6016ed7e8a25b6fc8f6234c5d37318d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -4px;\" \/>.<\/li>\n<li id=\"import-auto-id1498973\">Newton\u2019s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes. <\/li>\n<li id=\"import-auto-id1498977\">In symbols, Newton\u2019s second law of motion is defined to be\n<div data-type=\"equation\" class=\"equation\" id=\"eip-370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-898f31c44d5a1c71d79347a5cbe5acf5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b52c16cde0b8a68dd12fa28a3a4db6ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#70;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"32\" style=\"vertical-align: -3px;\" \/> is the net external force, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7161e60e68eaf90be5138c4a3040070d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: -3px;\" \/> is the change in momentum, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0714636704a254c71bede042781bc57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> is the change time.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1522174\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522179\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1466205\">An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522190\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1522191\">\n<p id=\"import-auto-id1466209\">An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522201\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1522202\">\n<p id=\"import-auto-id1466212\"><strong>Professional Application<\/strong><\/p>\n<p id=\"eip-id974177\">Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522215\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1522216\">\n<p id=\"import-auto-id1489046\"> How can a small force impart the same momentum to an object as a large force?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1522226\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1522230\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1522231\">\n<p id=\"import-auto-id1528696\">(a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2b2392e385c108952f64548b0cf0f892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/>. (b) Compare the elephant\u2019s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-53554bcfe1e143e59701f2eb439fbb0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -4px;\" \/>. (c) What is the momentum of the 90.0-kg hunter running at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae4137e8ee244dc5f417f106a134bfe5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/> after missing the elephant?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1493308\">\n<p id=\"import-auto-id1356784\">(a)<br \/>\n        <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7a4ffad0365b93d5e76c84d18f0f3838_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#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;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1489164\">(b) 625 to 1<\/p>\n<p id=\"import-auto-id1489166\">(c)<br \/>\n        <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-84d097336dcfeac5543e480b3993190e_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;&#54;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#50;&#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;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"123\" style=\"vertical-align: -4px;\" \/>\n    <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1493437\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1493438\">\n<p id=\"import-auto-id1486822\">(a) What is the mass of a large ship that has a momentum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-732b01073bbe6e662e02395aa4eba044_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;&#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;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/>, when the ship is moving at a speed of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a628b68f27dfadcfad915d533148209f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#56;&#46;&#48;&#32;&#107;&#109;&#47;&#104;&#63;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/> (b) Compare the ship\u2019s momentum to the momentum of a 1100-kg artillery shell fired at a speed of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-809e21302476d8c2108315fb21d6d905_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#48;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"71\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1493544\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1493545\">\n<p id=\"import-auto-id1466181\">(a) At what speed would a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e74be1b3de4caaabc7d3112883dd392_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;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -3px;\" \/> airplane have to fly to have a momentum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-732b01073bbe6e662e02395aa4eba044_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;&#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;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> (the same as the ship\u2019s momentum in the problem above)? (b) What is the plane\u2019s momentum when it is taking off at a speed of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5e2d3b2f0ff13850de94444944f265ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#46;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/>? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1493671\">\n<p id=\"import-auto-id1512421\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b8fc3f6748a807902186ff0b9812bfad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#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;&#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;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1528742\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ded14e7aceeebce1da7ddc4e2f58462d_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;&#50;&#48;&#125;&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;&#107;&#103;&#125;&middot;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1466163\">(c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the ship, the ship will not recoil very much. The recoil would be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44bc74f32131fb745b475bec6effec91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#48;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/>, which is probably not noticeable.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1493804\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1493805\">\n<p id=\"import-auto-id1489185\">(a) What is the momentum of a garbage truck that is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42cde23262656b7cc9212afe905a9838_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;&#50;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#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=\"78\" style=\"vertical-align: -3px;\" \/> and is moving at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b18bf6603242c887d1a95935c15ab3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -4px;\" \/>? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1493884\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1493885\">\n<p id=\"import-auto-id1488710\">A runaway train car that has a mass of 15,000 kg travels at a speed of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f4cf6fc2dfba6b06ea4d12c6dd52b2c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#52;&#32;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"60\" style=\"vertical-align: -4px;\" \/> down a track. Compute the time required for a force of 1500 N to bring the car to rest.\n<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1542630\">\n<p id=\"import-auto-id1488755\">54 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1542640\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1542641\">\n<p id=\"import-auto-id1488764\">The mass of Earth is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2533de5d10d448fa70380385f91fd52d_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;&#57;&#55;&#50;&#125;&times;&#123;&#49;&#48;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#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;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"95\" style=\"vertical-align: -3px;\" \/> and its orbital radius is an average of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6698058f687b50463f7a165352f39b39_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;&#57;&#54;&#125;&times;&#123;&#49;&#48;&#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;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -1px;\" \/>. Calculate its linear momentum.<\/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-id1499242\">\n<dt>linear momentum<\/dt>\n<dd id=\"fs-id1542740\">the product of mass and velocity<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1499250\">\n<dt>second law of motion<\/dt>\n<dd id=\"fs-id1542750\">physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes<\/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-421","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":417,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/421","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\/421\/revisions"}],"predecessor-version":[{"id":422,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/421\/revisions\/422"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/417"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/421\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=421"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=421"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=421"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=421"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}