Question

A ball of mass $4\text{kg}$ moving with velocity $4\text{m/s}$ collides with another identical ball at rest. The kinetic energy of the balls after collision is $\frac{7}{8}$ times of the original. Find the coefficient of restituton $\left(e\right)$.

• A. $1$
• B. $0.866$
• C. $0.732$
• D. $0.239$

Answer 1

The correct option is B $0.866$


Given:
$(K.E{)}_{final}=\frac{7}{8}(K.E{)}_{initial}$
$(K.E{)}_{loss}=(K.E{)}_{initial}-(K.E{)}_{final}$
$\Rightarrow (K.E{)}_{initial}-\frac{7}{8}(K.E{)}_{initial}$
$=\frac{1}{8}(K.E{)}_{initial}$ ...$\left(i\right)$

We know that
$(KE{)}_{loss}=\frac{1}{2}\times \frac{m_1{m}_2}{(m_1+m_2)}(u_{approach}{)}^2(1-e^2)$ ...$\left(ii\right)$
Equating $\left(i\right)$ and $\left(ii\right)$
$\frac{1}{2}\times \frac{m_1{m}_2}{(m_1+m_2)}\times \left(u_{approach}{)}^2\right(1-e^2)=\frac{1}{8}(K.E.{)}_{initial}$
$\Rightarrow \frac{1}{2}\times \left(\frac{m\times m}{m+m}\right)\times u_1^2\times (1-e^2)=\frac{1}{8}\left(\frac{1}{2}m{u}_1^2\right)$
$\Rightarrow \frac{1}{4}\times (1-e^2)=\frac{1}{8}\times \frac{1}{2}$
$\Rightarrow e^2=1-\frac{1}{4}=\frac{3}{4}$
or $e=\frac{\sqrt{3}}{2}=0.866$