Solve the following :
Question:

Consider a head-on collision between two particles of masses $m_{1}$ and $m_{2}$. The initial speeds of the particles are ${ }^{u_{1}}$ and ${ }^{u_{2}}$ in the same direction. The collision starts at $\mathrm{t}=0$ and the particles interact for a time interval $\Delta t$. During the collision, the speed of the first particle varies as $v(t)=u_{1}+\frac{t}{\Delta t}\left(v_{1}-u_{1}\right)$

Find the speed of the second particle as a function of time during the collision.

Solution:

Use C.O.L.M, $v_{1}(t)=V(t)$

$m_{1} u_{1}+m_{2} u_{2}=m_{1} V(t)+m_{2} V_{2}(t)$

$V_{2}(t)=\frac{m_{1} u_{1}+m_{2} u_{2}-m_{1} V(t)}{m_{2}}$

$=\frac{m_{1}}{m_{2}} u_{1}+u_{2}-\frac{m_{1}}{m_{2}} u_{1}-\frac{m_{1}}{m_{2}} \frac{t}{\Delta t}\left(v_{1}-u_{1}\right)$

$=\frac{m_{1}}{m_{2}} u_{1}+u_{2}-\frac{m_{1}}{m_{2}} u_{1}-\frac{m_{1}}{m_{2}} \frac{t}{\Delta t}\left(v_{1}-u_{1}\right)$

$=u_{2}-\frac{m_{1}}{m_{2}} \frac{t}{\Delta t}\left(v_{1}-u_{1}\right)$