Solution to "Another Cool Physics Problem"

The problem I posted a couple weeks ago really is quite cool. What makes an elementary physics problem cool? It think when it has one of two characteristics:

1) Either a simple application of basic principles renders a clear and elegant answer to an interesting problem, or

2) The problem requires an intuition that is not taught (perhaps it can't be) in the normal methodology of elementary physics. The normal approach we teach is: find the right equation and apply it, without much thought of what is actually happening. Cool problems, however, sometimes require a model.

The yo-yo problem found here is an example of the first type. The recent problem, found here, is of the second type.

One commenter suggested that the ball will reach it's initial height because of conservation of energy. That's a common guess, but it is wrong. Energy is conserved, but there is a lot of flexibility here: the height of each ball and the velocity of each ball.

If you have ever done this, you know that if you carefully drop the balls the smaller ball will bounce higher than its original height. But just using the equations you learn in elementary physics is a tricky approach--unless you have a model of the situation.

And here is the model:

1) The balls fall, together--as if they were attached. Computing the speed at impact is straightforward--conservation of energy shows that this speed, call it v_o, is the square root of 2gh.

2) Here is the gist of the solution: let's view what happens next this way: the larger ball elastically impacts the (approximately infinitely massive) ground and has its velocity instantaneously reversed--it becomes v_o directed up instead of down.

3) At that point we have an elastic collision between the big ball travelling upward at v_o and the small ball travelling downward at v_o. From conservation of momentum and energy, we can compute the recoil speed of the little ball.

4) From conservation of energy, we can then compute how high the little ball will go.

I'll leave it as an exercise that applying the equations for 1D elastic collisions in step 3. (Hint: the easiest way is to solve it in the frame where the little ball is travelling at 2v_o down, and the big ball is at rest, then transform back. That's because most text books will give you the equation for a 1D elastic collision when one object is at rest.) The answer: after the collision, the little ball will be travelling at 5v_o/3 upward.

The last step is simple: the little ball is travelling at 5v_o/3 upward starting at a height of 2R (the diameter of the big ball.) Simple conservation of energy shows it will reach a height of 2R + 25h/9, or 16h/9 above its initial height.