Question

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

• A. $2u$
• B. $\frac{3}{2}u$
• C. $\sqrt{5}u$
• D. $\sqrt{10}u$

Answer 1

The correct option is C

$\sqrt{5}u$

Step 1: Given Data:

Initial velocity of the train = $u$

Final velocity of the train = $3u$

Step 2: Parameter to be calculated:

The velocity with which the middle point of the train passes that same point

Step 3: Formula used:

$v^2-u^2=2aL$

Step 4: Calculation :

Let us consider the length of the train to be $L$.

The train is moving with constant acceleration.

The initial velocity is $u$, At the end velocity is $3u$.

Using $v^2-u^2=2aL$

$$\begin{gathered} 9u^2=u^2+2aL \\ 8u^2=2aL \end{gathered}$$

Hence $a=\frac{4u^2}{L}$

When the train has covered the L/2 distance; the velocity is

$$\begin{gathered} v^2=u^2+2\times \frac{4u^2}{L}\times \frac{L}{2} \\ {v}^2=u^2+4u^2 \\ v=\sqrt{5}u \end{gathered}$$

Thus, the middle point of the train passes $v=\sqrt{5}u$ through the same point.

Hence, option (C) is correct.