Question

$ A $ sphere $ S_{1} $ impings directly on an equal sphere $ S_{2} $ at rest. If the coefficient of restitution is $ e $ , then the velocities of $ S_{1} $ and $ S_{2} $ are in the ratio

• A. $ \frac{1+e}{1-e} $
• B. $ \frac{1-e}{1+e} $
• C. $ \frac{e-1}{e+1} $
• D. $ \frac{e+1}{e-1} $

Answer 1

Answer: (B)

Let $m$ be the mass of each sphere. Let $u$ be the velocity of the first sphere before impact and $v_{1}$ and $v_{2}$ be their velocities after impact, then
$v_{1}-v_{2}=-e u \, ....(i)$
and $m v_{1}+m v_{2}=m u$
$\Rightarrow v_{1}+v_{2}=u\, ....(ii)$
On solving Eqs. (i) and (ii), we get
$v_{1}=\frac{u(1-e)}{2}$ and $v_{2}=\frac{u(1+e)}{2}$
$\therefore \frac{v_{1}}{v_{2}}=\frac{1-e}{1+e}$