Question

An engine of a train, moving with uniform acceleration, passes the signal post with velocity $u$ and the last compartment passes the signal post with velocity $v$. The velocity with which the middle point of the train passes the signal post is :

• A. $\sqrt{\frac{v^2-u^2}{2}}$
• B. $\sqrt{\frac{v-u}{2}}$
• C. $\sqrt{\frac{v^2+u^2}{2}}$
• D. $\sqrt{\frac{v+u}{2}}$

Answer 1

The correct option is C $\sqrt{\frac{v^2+u^2}{2}}$

Suppose the length of the train is $l$.

So, $v^2-u^2=2al$
$\Rightarrow l=\frac{v^2-u^2}{2a}....\left(1\right)$

Also, let the speed of the middle point of the train is $v_m$.

Then, $v_m^2-u^2=2a\times \frac{l}{2}=al$

$\Rightarrow v_m^2-u^2=a\times \frac{v^2-u^2}{2a}$
$[$From $\left(1\right)]$

$\Rightarrow v_m=\sqrt{\frac{v^2+u^2}{2}}$

Hence, $\left(C\right)$ is the correct answer.