The first step in the problem-solving strategy asks you to clearly define the system. The goal is to choose a system whose momentum will be conserved. Even then, it is the total momentum of the system that is conserved, not the momenta of the individual objects within the system.

As an example, consider what happens if you drop a rubber ball and let it bounce off a hard floor. Is momentum conserved? You might be tempted to answer yes because the ball’s rebound speed is very nearly equal to its impact speed. But there are two errors in this reasoning.

First, momentum depends on velocity, not speed. The ball’s velocity and momentum vectors reverse direction during the collision. Even if their magnitudes are equal, the ball’s momentum after the collision is not equal to its momentum before the collision.

But more important, we haven’t defined the system. The momentum of what? Whether or not momentum is conserved depends on the system. FIGURE 11.15 shows two different choices of systems. In Figure 11.15a, where the ball itself is chosen as the system, the gravitational force of the earth on the ball is an external force. This force causes the ball to accelerate toward the earth, changing the ball’s momentum. When the ball hits, the force of the floor on the ball is also an external force. The impulse of $\vec{F}{}_{\text{floor on ball}}$ changes the ball’s momentum from “down” to “up” as the ball bounces. The momentum of this system is most definitely not conserved.

FIGURE 11.15 Whether or not momentum is conserved as a ball falls to earth depends on your choice of the system.

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Figure 11.15b shows a different choice. Here the system is $\text{ball}+\text{earth}$. Now the gravitational forces and the impulsive forces of the collision are interactions within the system. This is an isolated system, so the total momentum $\vec{P}=\vec{p}{}_{\text{ball}}+\vec{p}{}_{\text{earth}}$ is conserved.

In fact, the total momentum (in this reference frame) is $\vec{P}=\vec{0}$ because both the ball and the earth are initially at rest. The ball accelerates toward the earth after you release it, while the earth—due to Newton’s third law—accelerates toward the ball in such a way that their individual momenta are always equal but opposite.

Why don’t we notice the earth “leaping up” toward us each time we drop something? Because of the earth’s enormous mass relative to everyday objects—roughly ${10}^{25}$ times larger. Momentum is the product of mass and velocity, so the earth would need an “upward” speed of only about ${10}^{-25}\text{m/s}$ to match the momentum of a typical falling object. At that speed, it would take 300 million years for the earth to move the diameter of an atom! The earth does, indeed, have a momentum equal and opposite to that of the ball, but we’ll never notice it.