- #51

TSny

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##L_{N+1} = \frac{V_0}{v_N+V_N}L_1##.

Suppose ##\small M## essentially comes to rest at the ##\small N^{th}## collision. Then to a good approximation

##L_{N+1} = L_{N} = L_{closest}## and ##V_{N}=0##. So,

##L_{closest} = \frac{V_0}{v_N}L_1##. Conservation of energy implies ##\frac{1}{2}MV_0^2 = \frac{1}{2}mv_N^2##. So, ##L_{closest} = \sqrt{\frac{m}{M}}L_1##