Question

26. The two ends of a train moving with constant acceleration pass a certain point with velocities u and 3u. The velocity with which the middle point of the train passes the same point is
(1) 2u (2) $\frac{3}{2}u$ (3) $\sqrt{5}u$ (4) $\sqrt{10}u$

Answer 1

Dear student,
Let the length of the train be ‘L’.

As it crosses the point, the initial velocity can be taken as ‘u’ and final velocity as ‘v’.

So, v² = u² + 2aL

=> v² – u² = 2aL ……………(1)

Suppose the velocity with which the midpoint crosses that certain point be v^/. In this case the distance traveled is L/2.

So,

(v^/)² = u² + 2a(L/2)

=> (v^/)² = u² + ½ × (v² – u²) [using (1)]

=> (v^/)² = (v² + u²)/2

$=>v^{/}={\left(\frac{v^2+u^2}{2}\right)}^{\frac{1}{2}}$
Given u=u and v=3u so putting these value in above equation we get,
v^/ =$u\sqrt{5}$

This is the velocity of the midpoint of the train with which it crosses that point.

Regards