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A ball moving translationally collides elastically with another stationary ball of the same mass. At the moment of impact the angle between the straight line passing through the centres of the balls and the direction of the initial motion of the striking ball is equal to $α = 45^{\circ}$ Assuming the balls to be smooth. The fraction $η$ of the kinetic energy of the striking ball that turned into potential energy at the moment of the maximum deformation is ____ .

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answer is 0.25.

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Detailed Solution

Along line joining the centres, Apply LCLM under maximum deformation

$m_{1} u_{1} + m_{2} u_{2} = (m_{1} + m_{2}) V$

$m \frac{V}{\sqrt{2}} + 0 = 2 {mV}^{1} \Rightarrow V^{1} = \frac{V}{2 \sqrt{2}}$

At this stage K.E. of first mass

$= \frac{1}{2} m {(\frac{V}{2 \sqrt{2}})}^{2} + \frac{1}{2} m {(\frac{V}{\sqrt{2}})}^{2} = \frac{5}{8} (\frac{1}{2} {mV}^{2})$

$K.E. of second mass = \frac{1}{8} (\frac{1}{2} {MV}^{2})$

$Fraction converted to P.E. is \frac{2}{8} (\frac{1}{2} {mV}^{2}) = 0.25$

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