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 centers of the balls and the direction of the initial motion of the striking ball equal to $α = 45^{0}$ . Assuming the balls to be smooth, Find the fraction $η$ of the kinetic energy of the striking ball that turned into potential energy at the moment of the maximum deformation

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$0.5$

$0.75$

$0.25$

answer is B.

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Detailed Solution

Let the velocity of the colliding ball be $u$ . Applying the law of conservation of momentum along $x$ -axis, we have

$m u \cos (α) + 0 = m v_{1 x} + m v_{2 x}$

At the maximum deformation

$v_{1 x} = v_{2 x} v_{1 x} = \frac{u \cos α}{2}$

Initial K.E. of the system

$= \frac{1}{2} m u^{2}$

Final K.E. of the system

$= \frac{1}{2} m {v_{1 x}}^{2} + \frac{1}{2} m {v_{2 x}}^{2} = \frac{m u^{2} \cos^{2} α}{4}$

now $η = \frac{[\frac{m u^{2} \cos^{2} α}{4}]}{\frac{1}{2} m u^{2}}$ =0.25

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