Question

A sphere impinges directly on a similar sphere at rest. If the coefficient of restitution is $\frac{1}{2}$, the velocities after impact are in the ratio

• A. 1 : 2
• B. 2 : 3
• C. 1 : 3
• D. 3 : 4

Answer 1

Answer: (C)

1 : 3

Answer 2

From principle of conservation of momentum ,

$mv_{1} + m^{'}v_{1}^{'} = mu_{1} + m^{'}u_{1}^{'}$ ......(i)

From Newton's rule for relative velocities before and after impact.

$v_{1} - v_{1}^{'} = - e(u_{1} - u_{1}^{'})$ ......(ii)

Here $m = m^{'}$, $e = \frac{1}{2}$ and let $u_{1}^{'} = 0$

Then from (i) $mv_{1} + mv_{1}^{'} = mu_{1}$ or $v_{1} + v_{1}^{'} = u_{1}$ ......(iii)

From (ii), $v_{1} - v_{1}^{'} = \frac{- 1}{2}u_{1}$ .......(iv)

Adding (iii) and (iv) , $2v_{1} = \frac{1}{2}u_{1}$ or $R = u\sqrt{\frac{2h}{g}} = (u\cos\theta) \times t$ ......(v)

From (iii) and (v), $v_{1}^{'} = u_{1} - \frac{1}{4}u_{1} = \frac{3}{4}u_{1}$. Hence $v_{1}:v_{1}^{'} = 1:3$.