Question

A stationary particle explodes into two particles of masses $m_1$ and $m_2$ which move in opposite directions with velocities $v_1$ and $v_2$. The ratio of their kinetic energies $E_1 /E_2$ is

• A. $1$
• B. $ m_{1}v_{2}/m_{2}v_{1} $
• C. $ m_{2}/m_{1} $
• D. $ m_{1}/m_{2} $

Answer 1

The Correct Option is C

For a exploding body, linear momentum is conserved.
From conservation of linear momentum
$P_{\text {initial }}=P_{\text {final }} $
$0=m_{1} v_{1}-m_{2} v_{2} $
or $m_{1} v_{1}=m_{2} v_{2} $
or $ \frac{v_{1}}{v_{2}}=\frac{m_{2}}{m_{1}}\,\,\,$...(i)
Thus, ratio of kinetic energies
$\frac{E_{1}}{E_{2}}=\frac{\frac{1}{2} m_{1} v_{1}^{2}}{\frac{1}{2} m_{2} v_{2}^{2}}=\frac{m_{1}}{m_{2}} \times\left(\frac{m_{2}}{m_{1}}\right)^{2} $
$=\frac{m_{2}}{m_{1}}$