Question

A smooth small spherical ball of mass m, moving with velocity u collides head on with another small spherical ball of mass m, which was initially at rest. One-third of the initial kinetic energy of the system is lost. The coefficient of restitution between the spheres is?

• A. $\frac{1}{3}$
• B. $\frac{1}{\sqrt{3}}$
• C. $\frac{1}{2}$
• D. Zero

Answer 1

The correct option is B $\frac{1}{\sqrt{3}}$

Given : $u_1=u$ $u_2=0$ $m_1=m_2=m$

Initial kinetic energy of system $K.E_i=\frac{1}{2}m{u}^2$

Loss in kinetic energy is given by $\Delta K.E=\frac{1}{2}\left(\frac{m_1{m}_2}{m_1+m_2}\right)(1-e^2)(u_1-u_2{)}^2$

$\therefore$ $\frac{1}{3}\times \frac{1}{2}m{u}^2=\frac{1}{2}\left(\frac{m^2}{2m}\right)(1-e^2)(u-0{)}^2$

Or $1-e^2=\frac{2}{3}$

Or $e^2=\frac{1}{3}$

$⟹e=\frac{1}{\sqrt{3}}$