Question

A sphere of mass $m$ moving with a constant velocity $u$ hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution, then the ratio of the velocity of two spheres after collision will be

• A. $\frac{1-e}{1+e}$
• B. $\frac{1+e}{1-e}$
• C. $\frac{e+1}{e-1}$
• D. $\frac{e-1}{e+1}{t}^2$

Answer 1

Answer: (A)

$\frac{1-e}{1+e}$

Using conservation of momentum

$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$

$mu+0=m(v_1+v_2)$

$v_1+v_2=u\dots ..\left(I\right)$

We know that,

The coefflclent of restltutlon Is

$e=\frac{\text{velocity of seperation}}{\text{velocity of approach}}$

$e=\frac{v_2-v_1}{u_1-u_2}$

$e=\frac{v_2-v_1}{u}$

$v_2-v_1=eu\dots ..\left(\text{II}\right)$

From equation (I) and (II)

$2v_2=u+ue$

$v_2=\frac{u(1+e)}{2}\dots ..\left(III\right)$

Now, put the value of $v_2$ in equation (I)

$v_1+\frac{u(1+e)}{2}=u$

$v_1=\frac{u(1-e)}{2}$

The ratio of the velocity of two sphere will be

$\frac{v_1}{v_2}=\frac{\frac{u(1-e)}{2}}{\frac{u(1+e)}{2}}$

$\frac{v_1}{v_2}=\frac{1-e}{1+e}$