Question

A sphere of mass m moving with a constant velocity hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution, then ratio of velocities of the 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. $\frac{(e+1)}{(e-1)}$
  

Answer 1

The correct option is A $\frac{(1-e)}{(1+e)}$

Given : $m_1=m_2=m$

Velocity of moving sphere (first) before collision $u_1=v$

Velocity of stationary sphere (second) before collision $u_2=0$

Velocity of first sphere after collision $v_1=\frac{(m_1-e{m}_2)u_1+(1+e)m_2{u}_2}{m_1+m_2}$

$⟹$ $v_1=\frac{(m-em)v+(1+e)m\left(0\right)}{m+m}=\frac{(1-e)v}{2}$

Velocity of second sphere after collision $v_2=\frac{(m_2-e{m}_1)u_2+(1+e)m_1{u}_1}{m_1+m_2}$

$⟹$ $v_2=\frac{(m-em)\left(0\right)+(1+e)mv}{m+m}=\frac{(1+e)v}{2}$

Thus we get ratio of velocities $\frac{v_1}{v_2}=\frac{1-e}{1+e}$