The "galilean cannon" is a stack of elastic balls that are dropped from a height onto a flat surface. when they rebound, the smallest ball at the top can bounce to incredible heights. imagine a stack of n balls with masses m, , 2m, 3m, … etc. where > 1. the balls are dropped from a height ℎ.

a. assume all the collisions between balls are elastic. derive an expression for the height that the smallest ball bounces to, using m, ℎ, or some subset thereof

b. now assume that the ball-ball collisions are inelastic; when 2 balls collide, their total kinetic energy decreases by some factor i. e. = , with < 1. find such that the topmost ball rises back to the initial height ℎ. assume the bottom ball’s collision with the ground is perfectly elastic.