Question

A sphere of mass $m$ moving with constant velocity $ u $ , collides with another stationary sphere of same mass. If $e$ is the coefficient of restitution, the ratio of the final velocities of the first and second spheres is

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

Answer 1

Answer: (B)

Let $v_{1}, v_{2}$ be the final velocities of the two spheres.
Applying the law of conservation of linear momentum
$m u=m\left(v_{1}+v_{2}\right)$
or $v_{1}+v_{2}=u \ldots$ (i)
Again the coefficient of restitution is given by
$e=\frac{v_{2}-v_{1}}{u}$
or $v_{1}+v_{2}=u \ldots$ (ii)
Solving Eqs. (i) and (ii), we get
$v_{1}=\frac{u}{2}(1-e), v_{2}=\frac{u}{2}(1+e)$
Therefore, $\frac{v_{1}}{v_{2}}=\left(\frac{1-e}{1+e}\right)$