Conservation of Kinetic Energy and Linear Momentum In an Elastic Collision

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The Idea

When one object strikes another in such a way that the objects bounce off each other, the collision is said to be elastic. When this happens, whatever momentum you start off with, you have at the end. In the case of an elastic collision, the objects also move off with the same overall kinetic energy they started with. In this project, we explore what happens when collisions are elastic.

What You Need

• 4 pool balls (or hardballs or golf balls)
• track for the balls to roll in one dimension (This can easily be set up by taping 2 meters sticks to a smooth board)
• large ball, such as a basketball
• smaller ball, such as a ping-pong ball
• optional—a Newton's cradle, as shown in Figure 52-1

One ball hitting three

1. Place three balls of equal mass in the track.
2. Place the fourth ball a few inches away in the track.
3. Roll the ball, so it collides with the other three.
4. (There are several other ways to do this, including a Newton's cradle or four equal mass sliders in an air track.)

Big ball/small ball

1. Place the small ball on top of the large ball.
2. Drop both balls together. (Caution: do this in a place where, if the small ball goes flying off, it won't break anything and won't hurt anyone. If the balls you are using are small enough, you may be able to do this in a clear plastic vertical guide or in a large graduated cylinder.)

One ball hitting four

The incoming ball comes to a dead stop, as shown in Figures 52-2 and 52-3. The outermost stationary ball moves in the same direction and at the same velocity as the incoming ball. The other three stationary balls do not move.

Big ball/small ball

The balls bounce together. After striking the ground, the smaller ball flies off with much greater velocity than the large ball.

Why It Works

With the stack of balls, it is not hard to understand how the momentum of the incoming ball is transferred to the ball that gets knocked out of the stack. This is a clear illustration of conservation of linear momentum.

But why is only one ball knocked out of the stack? Why, for instance, do we never have two balls knocked out with each taking one half of the momentum of the incoming ball? That would also be perfectly consistent with the law of conservation of momentum. The problem is these collisions are elastic collisions, which means not only is momentum conserved, but kinetic energy is also conserved. The only way this can happen is for a single ball to emerge from the stack with the same momentum as the incoming ball.

With the large and small balls, the large ball having a larger mass conserves momentum by causing the smaller ball with a lower mass to fly off with a larger velocity.

Other Things to Try

A Newton's cradle, as shown in the previous Figure 52-1, is another good way to study elastic collisions. In a Newton's cradle, two balls never rebound when struck by a single ball and three balls never rebound when struck by two balls. This is the result of both conservation of linear momentum and conservation of energy.

The Point

In an elastic collision, both linear momentum and kinetic energy are conserved.

When momentum is transferred from one object to another, a larger velocity compensates for a smaller mass. In the case of an elastic collision between objects of equal mass, this condition can be met only when the same number of balls move after the collision as were moving before.