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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