# The Physics Philes, lesson 36: Knock’em About

In which kinetic energy is conserved, algebra is done, and notes are remembered.

Woo hoo! I remembered to bring my notes and textbook home from work, which means that we can continue our discussion of momentum! Huzzah!

If you’ll remember, an elastic collision in an isolated system is a collision in which both kinetic energy and momentum are conserved. You get an elastic collision when the forces between the colliding objects are conserved. So, kinetic energy in this type of collision looks like this:

As always, conservation of momentum looks like this:

OK, so that’s a lot of letters. Think of billiard balls. when they smash together, a little kinetic energy is stored as potential energy for a bit, but this is reconverted into kinetic energy. See? It’s really not so hard to understand.

Let’s talk for a second about what happens in an elastic collision when one of the objects is at rest. If we just use the above equations, we can solve for the final velocities of both objects, and the initial velocity of moving object. First, we need to rearrange the conservation of kinetic energy and the conservation of momentum equations:

If we divide the first equation by the second, we get:

Now let’s substitute the resulting equation into the rearranged conservation of kinetic energy equation:

If we substitute this fraction into the previous quotient of the rearranged conservation of kinetic energy and momentum equations, we get:

Ok, ok, ok. Enough math. What does it mean? Think about a ping pong ball hitting an at rest bowling ball. We know from living in the world that the ping pong ball is probably not going to move the bowling ball. At least, not very much. We’d expect the ping pong ball to bounce back at pretty much the same speed as it had when it was traveling toward the bowling ball. That’s all this series of equations is predicting.

But what happens when the masses of the two objects are close to the same? In that case, the moving body stops and transfers all its momentum and kinetic energy to body that wasn’t moving. That’s why billiard balls work the way they do.

That’s kind of a special case. Let’s go back to what happens when there is a difference in the masses of the two objects. In that case, the velocity of B relative to the velocity of A after the collision is:

Velocity vA1x is the negative of the velocity of B relative to A before the collision. (Relative velocity has the same magnitude but the opposite sign.) This is because before the collision, the objects were moving toward each other; after the collision, they are moving apart. This is true for any straight line elastic collision, regardless of whether one of the bodies were at rest or not. What does this mean? It means we have another definition for elastic collisions! In an elastic collision, the relative velocity of the two objects has the same magnitude before and after the collision. Whenever this element is satisfied, we know kinetic energy is conserved.

Alrighty, ladies and gentlemen. That’s all the physics I have in me for today. If you will excuse me, I need to go study for the trig exam I have on Tuesday. With me luck!

Featured image credit: S. Parker

### 2 Comments

1. I forgot how neat this stuff was. I’m impressed that you were able to condense elastic collisions into one blog post.

By the way, you forgot to put a subscript underneath an m…

• This stuff is neat! But I think the last few of posts have had something to do with elastic collisions, so I haven’t really managed to smash the topic into one post. (If only I were that good!)

Thanks for letting me know about the subscript. I’ll fix it as soon as I’m able.