The question as it appears in the 03/06 edition of Specs & Techs from GlobalSpec:
A very small ball is on the top of another ball with a radius of 5 cm; the mass of the smaller ball is negligible compared to the larger ball. The balls are dropped from a distance of 50 cm between the ground and the bottom of the larger ball. Lying on a roof 4.5 meters from the ground, you observe this experiment looking directly to the falling balls. If all collisions are elastic, will you be hit in the face when the balls bounce? If so, by which ball?
(Update 4:55 PM EST 03/12/07) And the Answer is....
A moment before the big ball hits the ground both balls are speeding downward with a velocity v given by
where g is the acceleration of gravity, and h is the original height (50 cm).
(See the given figure).
Immediately after the big ball hits the ground, it moves upward with speed v. The small ball, however, is still moving downward at speed v. Therefore, the relative speed of the two balls is 2v. After the balls bounce off each other, the upward speed of the big ball stays equal to v (because its mass is much bigger than the mass of the small ball), and the upward speed of the small ball is
By applying the conservation of energy to each ball we get:
1) For the big ball
It is clear from this equation that the height reached by the big ball is the same as the original height (50 cm).
2) For the small ball. The conservation of energy equation is given by
where H is the height reached after bouncing off the big ball. From this equation we find that
or
Then, the maximum height reached by the small ball is
This is depicted in the figure.
Certainly, if you don't move fast you will be hit in the face by the small ball.

Re: Bouncing Balls: Newsletter Challenge (03/06/07)