Question

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

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

Answer 1

Answer: (C)

$\sqrt{(u^2+v^2)/2}$

Let $s$ be the length of train using $v{}^2=u{}^2+2a\times s$ $or2as=v^2-u^2$ ......(1)
Where $v$ is final velocity after travelling a distance $s$ with an acceleration $a$ and $u$ is initial velocity as per question
Let velocity of middle point of train at same point is $v{}^{\prime }$, then
$\left(v{}^{\prime }\right){}^2=u{}^2+2a\times \left(s/2\right)$ $=u^2+as$ .....(2)
By equations (1) and (2), we get $v^{\prime }=\sqrt{\frac{v{}^2+u{}^2}{2}}$