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There are five balls at rest at equal distances in a straight line on a smooth horizontal surface. Their masses are in geometrical progression with common ratio 2 and their coefficients of restitution are each 2/3. If the first ball is started towards the second with velocity u, then the velocity communicated to 5th ball is $u {(\frac{x}{y})}^{(y - x)} .$ Find the value of $x + y$

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answer is 14.

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Detailed Solution

We know that velocity of 2nd ball after collision is given by $υ_{2} = \frac{u_{1} (1 + e) m_{1}}{(m_{1} + m_{2})} + u_{2} \frac{(m_{2} - m_{1})}{(m_{1} + m_{2})}$

In present problem $u_{2} {= 0, m}_{2} {= 2m}_{1} and e = 2/3,$ hence $υ_{2} = \frac{u (1 + \frac{2}{3})}{(m_{1} + 2 m_{1})} = \frac{5}{9} u$

As four exactly similar type of collisions are taking place successively,

hence velocity communicated to fifth ball $υ_{5} = {(\frac{5}{9})}^{4} u$

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