# Basketball and tennis ball: Part II

Classical Mechanics Level pending

Now consider n balls, $$B_{1}, \cdots , B_{n}$$ having masses $$m_{1}, m_{2}, \cdots , m_{n}$$ (with $$m_{1} >>m_{2}>> \cdots >> m_{n}$$ ) , sitting in a vertical stack. The bottom of $$B_{1}$$ is a height h above the ground, and the bottom of $$B_{n}$$ is a height $$h + l$$ above the ground. The balls are dropped. In terms of n, to what height does the top ball bounce?

Note: Work in the approximation where $$m_{1}$$ is much larger than $$m_{2}$$, which is much larger than $$m_{3}$$, etc., and assume that the balls bounce elastically. If $$h = 1$$ meter, what is the minimum number of balls needed for the top one to bounce to a height of at least 1 kilometer? To reach escape velocity? Assume that the balls still bounce elastically (which is a bit absurd here). Ignore wind resistance, etc., and assume that l is negligible.

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