Question

The two ends of a train moving with uniform acceleration pass a certain point with velocities $uandv$. The velocity with which the middle point of the train passes the same point is:

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

Answer 1

Answer: (A)

Since the train has uniform acceleration, we know,

$v^2=u^2+2al$ ( $l$=length of train ),
so the acceleration of the train can be found out using this, so, a $=\frac{v^2-u^2}{2l}$.

For the central point to find the velocity:

$s$ or the displacement will be half the length of the train that is '$l$' at the midpoint and velocity at the midpoint will be $v_m$.
So, $v_m^2=u^2+2a\frac{l}{2}\Rightarrow v_m^2-u^2=al$
$v_m^2-u^2=\frac{v^2-u^2}{2}\Rightarrow v_m^2=\frac{v^2+u^2}{2}$
$v_m=\sqrt{\frac{v^2+u^2}{2}}$