# Momentum

## Conservation of Momentum


### Section Key Terms


### Conservation of Momentum

It is important we realize that momentum is conserved during collisions, explosions, and other events involving objects in motion. To say that a quantity is conserved means that it is constant throughout the event. In the case of conservation of momentum, the total momentum in the system remains the same before and after the collision.
You may have noticed that momentum was not conserved in some of the examples previously presented in this chapter. where forces acting on the objects produced large changes in momentum. Why is this? The systems of interest considered in those problems were not inclusive enough. If the systems were expanded to include more objects, then momentum would in fact be conserved in those sample problems. It is always possible to find a larger system where momentum is conserved, even though momentum changes for individual objects within the system.
For example, if a football player runs into the goalpost in the end zone, a force will cause him to bounce backward. His momentum is obviously greatly changed, and considering only the football player, we would find that momentum is not conserved. However, the system can be expanded to contain the entire Earth. Surprisingly, Earth also recoils—conserving momentum—because of the force applied to it through the goalpost. The effect on Earth is not noticeable because it is so much more massive than the player, but the effect is real.
Next, consider what happens if the masses of two colliding objects are more similar than the masses of a football player and Earth—in the example shown in  of one car bumping into another. 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, assuming that the effects due to friction are negligible. Car m1 slows down as a result of the collision, losing some momentum, while car m2 speeds up and gains some momentum. If we choose the system to include both cars and assume that friction is negligible, then the momentum of the two-car system should remain constant. Now we will prove that the total momentum of the two-car system does in fact remain constant, and is therefore conserved.
Using the impulse-momentum theorem, the change in momentum of car 1 is given by
where F1 is the force on car 1 due to car 2, and  
  is the time the force acts, or the duration of the collision.
Similarly, the change in momentum of car 2 is  
  where F2 is the force on car 2 due to car 1, and we assume the duration of the collision  
  is the same for both cars. We know from Newton’s third law of motion that F2 = –F1, and so  
 .
Therefore, 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 p′1 and p′2 are the momenta of cars 1 and 2 after the collision.
This result that momentum is conserved is true not only for this example involving the two cars, but for any system where the net external force is zero, which is known as an isolated system. The law of conservation of momentum states that for an isolated system with any number of objects in it, the total momentum is conserved. In equation form, the law of conservation of momentum for an isolated system is written as 
or
where ptot is the total momentum, or the sum of the momenta of the individual objects in the system at a given time, and p′tot is the total momentum some time later.
The conservation of momentum principle can be applied to systems as diverse as a comet striking the Earth or a gas containing huge numbers of atoms and molecules. Conservation of momentum appears to be 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.

### Check Your Understanding


### Section Summary

1. The law of conservation of momentum is written ptot = constant or ptot = p′tot (isolated system), where ptot is the initial total momentum and p′tot is the total momentum some time later.
2. In an isolated system, the net external force is zero.
3. Conservation of momentum applies only when the net external force is zero, within the defined system.

### Key Equations


### Concept Items


### Critical Thinking


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
