Question

A sphere of mass $m$ moving with a constant velocity $v$ hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution, then the ratio of the velocities of the first sphere to the second spheres after the collision will be :

• A. $\left(\frac{1+e}{1-e}\right)$
• B. $\left(\frac{e-1}{e+1}\right)$
• C. $\left(\frac{1-e}{e+1}\right)$
• D. $\left(\frac{1+e}{e-1}\right)$

Answer 1

The correct option is C $\left(\frac{1-e}{e+1}\right)$

Here, a sphere of mass $m$ moving with a constant velocity $v$ hits another stationary sphere of the same mass. As per the given problem we can write
$v_1-v_2=-ev$ ..................(1)
and
$v_1+v_2=v$ ..................(2)
Adding (1) and (2),
$2v_1=v(1-e)$
$v_1=\frac{(1-e)}{2}v$
Using this in (2). we get
$v_2=\frac{(e+1)}{2}v$
Therefore,
$\frac{v_1}{v_2}=\frac{\frac{(1-e)}{2}v}{\frac{(e+1)}{2}v}$
$\frac{v_1}{v_2}=\frac{1-e}{e+1}$