Momentum is an important quantity because it is conserved. Yet it was not conserved in the examples in previous chapters where large changes in momentum were produced by forces acting on the system of interest. Under what circumstances is momentum conserved?

The answer to this question entails considering a sufficiently large system. It is always possible to find a larger system in which total momentum is constant, even if momentum changes for components of the system. If a football player runs into the goalpost in the end zone, there will be a force on him that causes him to bounce backward. However, the Earth also recoils —conserving momentum—because of the force applied to it through the goalpost. Because Earth is many orders of magnitude more massive than the player, its recoil is immeasurably small and can be neglected in any practical sense, but it is real nevertheless.

Consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—for example, one car bumping into another, as shown in Figure 1. Both cars are coasting in the same direction when the lead car (labeled m₂) is bumped by the trailing car (labeled m₁). The only unbalanced force on each car is the force of the collision. (Assume that the effects due to friction are negligible.) Car 1 slows down as a result of the collision, losing some momentum, while car 2 speeds up and gains some momentum. We shall now show that the total momentum of the two-car system remains constant.

Using the definition of impulse, the change in momentum of car 1 is given by

where [latex]\boldsymbol{\vec{\textbf{F}}_1}[/latex] is the force on car 1 due to car 2, and Δt is the time the force acts (the duration of the collision). Intuitively, it seems obvious that the collision time is the same for both cars, but it is only true for objects traveling at ordinary speeds. This assumption must be modified for objects travelling near the speed of light, without affecting the result that momentum is conserved.

Similarly, the change in momentum of car 2 is

where [latex]\boldsymbol{\vec{\textbf{F}}_2}[/latex] is the force on car 2 due to car 1, and we assume the duration of the collision Δt is the same for both cars. We know from Newton’s third law that [latex]\boldsymbol{\vec{\textbf{F}}_1=-\vec{\textbf{F}}_2}[/latex], and so

Thus, the changes in momentum are equal and opposite, and

Because the changes in momentum add to zero, the total momentum of the two-car system is constant. That is,

where [latex]\boldsymbol{\vec{\textbf{p}}^{\prime}_1}[/latex] and [latex]\boldsymbol{\vec{\textbf{p}}^{\prime}_2}[/latex] are the momenta of cars 1 and 2 after the collision. (We often use primes to denote the final state.)

This result—that momentum is conserved—has validity far beyond the preceding one-dimensional case. It can be similarly shown that total momentum is conserved for any isolated system, with any number of objects in it. In equation form, the conservation of momentum principle for an isolated system is written

or

where [latex]\vec{\textbf{p}}_{\textbf{tot}}[/latex] is the total momentum (the sum of the momenta of the individual objects in the system) and [latex]\boldsymbol{\vec{\textbf{p}}^{\prime}_{\textbf{tot}}}[/latex] is the total momentum some time later. (The total momentum can be shown to be the momentum of the center of mass of the system.) An isolated system is defined to be one for which the net external force is zero [latex]\boldsymbol{(\vec{\textbf{F}}_{\textbf{net}}=0).}[/latex]

Perhaps an easier way to see that momentum is conserved for an isolated system is to consider Newton’s second law in terms of momentum, [latex]\boldsymbol{\vec{\textbf{F}}_{\textbf{net}}=\frac{\Delta\vec{\textbf{p}}_{\textbf{tot}}}{\Delta{t}}}.[/latex] For an isolated system, [latex]\boldsymbol{\vec{\textbf{F}}_{\textbf{net}}=0};[/latex] thus, [latex]\boldsymbol{\Delta\vec{\textbf{p}}_{\textbf{tot}}=0},[/latex] and [latex]\vec{\textbf{p}}_{\textbf{tot}}[/latex] is constant.

We have noted that the three length dimensions in nature—x, y, and z—are independent, and it is interesting to note that momentum can be conserved in different ways along each dimension. For example, during projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero and momentum is unchanged. But along the vertical direction, the net vertical force is not zero and the momentum of the projectile is not conserved. (See Figure 2.) However, if the momentum of the projectile-Earth system is considered in the vertical direction, we find that the total momentum is conserved.

 Conservation of momentum is violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force. For example, in the collision of two cars considered above, the two-car system conserves momentum while each one-car system does not.