Question

A sphere ${}^{\prime }{P}^{\prime }$ of mass ${}^{\prime }{m}^{\prime }$ moving with velocity ${}^{\prime }{u}^{\prime }$ collides head-on with another sphere ${}^{\prime }{Q}^{\prime }$ of mass ${}^{\prime }{m}^{\prime }$ which is at rest. The ratio of final velocity of ${}^{\prime }{Q}^{\prime }$ to initial velocity of ${}^{\prime }{P}^{\prime }$ is
($e=$ coefficient of restitution)

• A. $\frac{e-1}{2}$
• B. ${\left[\frac{e+1}{2}\right]}^{1/2}$
• C. $\frac{e+1}{2}$
• D. ${\left[\frac{e+1}{2}\right]}^2$

Answer 1

The correct option is C $\frac{e+1}{2}$

Here, $m_1=m_2=m$, $u_1=u$, $u_2=0$
Let $v_1,v_2$ be their velocities after collision.
According to principle of conservation of linear momentum
$mu+0=m(v_1+v_2)$
or $v_1+v_2=v$ ....(i)
By definition, $e=\frac{v_2-v_1}{u-0}$
or $v_2-v_1=eu$ .....(ii)
Adding equations (i) and (ii), we get
$v_2=\frac{u(1+e)}{2}\Rightarrow \frac{v_2}{u}=\frac{1+e}{2}$