Stadium Dugout Announcer's Booth Batting Cage Bullpen Behind the Plate

## Momentum

When the bat hits the ball, it exerts some force on the ball. However, does this force account for the distance the ball travels?

Let's think about it. Just imagine a home run hitter like Babe Ruth hitting a stationary ball. How far do you think will it go? Will it go more than four hundred feet? Probably not. While the kinetic energy transferred from the bat to the ball accounts for some of the energy of the ball, it does not account for all. So where is this mysterious energy coming from?

The answer is conservation of momentum.

What is momentum? Momentum is a vector describing a "quantity of motion" or in mathematical terms p (momentum) = mass * velocity.

I just said that momentum is conserved (almost, read more to find out why) but how do I know that? I know that because of Newton's 2nd law: F=ma (Force equals mass times acceleration).

As was explained in the position page, acceleration is dv/dt, which is the change in the velocity (dv) divided by the change in time (dt). "Change in velocity?" you ask? Well, the change velocity (dv) is the difference between the current value and the last value of the velocity. For our purposes, we can say that dv=vf-vi, or, the change in velocity is equal to the final velocity minus the initial velocity. We also know that in a system with no external forces, the total force, F, is zero. This is because if you push against a wall, that wall pushes back against you with the same amount of force. So, assuming there are no external forces and plugging dv/dt for a into F=ma:

F = m dv/dt = 0
Multiply both sides by dt
m dv = 0
Now plug in vf-vi for dv
m vf - m vi = 0
the mass times the final velocity minus the mass times the initial velocity equals zero

From that last equation, it tells us that for the same change in time (dt), the difference between the final momentum and the initial momentum always remains the same.

Okay lets think about this more. Conservation of momentum means that harder you throw the harder the ball will bounce back at you. Just think about throwing a ball against a solid wall. The harder you throw the ball against the wall, the harder it bounces back. That is the reason it is easier to hit a home run on a fastball than on a curveball.

Conservation of momentum also means that the bat can transfer some of its momentum to the ball. This is why it can be better to use a heavier bat if you can swing it just as fast. The momentum is the product of the mass and the velocity, so a heavier bat swung with the same speed as a lighter bat will have more momentum.

Now, you may have noticed that I said momentum is "almost" conserved. Why isn't it if the equations say it should be? Well, momentum is always conserved in a closed system, but a baseball game in the real world is not a closed system. The bat and the ball are elastic materials. When the ball hits the bat, the ball will be squished to a certain degree. After few milli-seconds, it rebounds back. This contraction and rebound action is caused by the heat, or energy generated by friction, and some momentum is lost, or transferred elsewhere, in this action. There are also other factors that can use up energy, however, the concept of conservation of momentum is still relevant in predicting the range of a baseball.