Question

Two particles of equal mass m have respective initial velocities $u\hat{i}$ and $u\left(\frac{\hat{i}+\hat{j}}{2}\right).$ They collide completely inelastically. The energy lost in the process is :

• A. $\frac{3}{4} mu^{2}$
• B. $\sqrt{\frac{2}{3}} mu^{2}$
• C. $\frac{1}{3} mu^{2}$
• D. $\frac{1}{8} mu^{2}$

Answer 1

Answer: (D)

From momentum conservation
$mu\hat{i}+mu\left(\frac{\hat{i}+\hat{j}}{2}\right)=\left(m+m\right)\bar{v}$
$\Rightarrow \bar{v}=\frac{3}{4}u\hat{i}+\frac{u}{4}\hat{j}$
$\Rightarrow \left|v\right|=\frac{u}{4}\sqrt{10}$
Final kinetic energy $=\frac{1}{2}2m\left(\frac{u}{4}\sqrt{10}\right)^{2}=\frac{5}{8}mu^{2}$ Initial kinetic energy
$=\frac{1}{2}mu^{2}+\frac{1}{2}m\left(\frac{u}{\sqrt{2}}\right)^{2}=\frac{6}{8}mu^{2}$
Loss in $K.E.=k_{i}-k_{f}=\frac{1}{8}mu^{2}$