Solving Elastic Collision Problems

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Changing frames doesn't change the amount of internal energy created (it only changes the mechanical energy we see), so having the objects stick together results in the largest possible creation of internal energy.

If we are told that a given collision is elastic (or at least can be approximated as such), then that gives us an additional condition that we can use to solve the problem. in each case, the diagram will show the experimental result, which we will then show mathematically using the combination of momentum and kinetic energy conservation.

Intuitively we know we would rather be in the heavier vehicle, but why?

Well, we would want to experience as little force as possible (force is what breaks bones).

This of course does not mean that all of the kinetic energy is lost (the objects do continue moving at the end in most such collisions), only that they don't bounce off each other.

From the perspective of the center of mass frame, we can see that such a collision maximizes the amount of internal energy that the collision can create: In this frame, the objects stop entirely after the collision, so all of the mechanical energy becomes internal.

Clearly we want the bowling ball to have more mass than a pin, so that it can carry through to the pins behind the front pin(s).

If we consider collisions in two dimensions (which we will do later), we will find that the angular deflection of the ball when it doesn't strike the pin head-on will be less when the ball is heavier, which is one reason heavier bowling balls are more effective than lighter ones.

As a second example of this, suppose we are passengers in one of two vehicles involved in a head-on collision.

Which vehicle would we rather be in, the lighter one or the heavier one?


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