Question

The two ends of a train moving with a constant acceleration pass a certain pole with velocities $u$ and $v$. The velocity with which the middle point of the train passes the same pole is

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

Answer 1

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

Let the length of the train $=l$ and uniform acceleration $=f$

$\therefore v^2=u^2+2fl$

$\Rightarrow f=\frac{v^2-u^2}{2l}$

Let the desired speed be $v^{\prime }$. The train will have moved $\frac{l}{2}$ till it attains a speed $v^{\prime }$.

$v^{\prime 2}=u^2+2f\cdot \frac{l}{2}=u^2+fl$

$=u^2+\frac{v^2-u^2}{2l}\cdot l$

$=u^2+\frac{v^2-u^2}{2}$

$=\frac{u^2+v^2}{2}$

$v^{\prime }=\sqrt{\frac{u^2+v^2}{2l}}$