A goods train accelerating uniformly on a straight railway track, approaches an electric pole standing on the side of track. Its engine passes the pole with velocity u and the guard’s room passes with velocity v. The middle wagon of the train passes the pole with a velocity

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$\sqrt{uv}$

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

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

$\frac{1}{2} \sqrt{u^{2} + v^{2}}$

answer is D.

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Detailed Solution

Let ‘S’ be the distance between two ends and ‘a’ be the constant acceleration

As we know $v^{2} - u^{2} = 2 aS$

$or, aS = \frac{v^{2} - u^{2}}{2}$

Let v_c be velocity at mid point.

Therefore, $v_{c}^{2} - u^{2} = 2 a \frac{S}{2}$

$\begin{array}{l} v_{c}^{2} = u^{2} + aS \\ v_{c}^{2} = u^{2} + aS \\ v_{c} = \sqrt{\frac{u^{2} + v^{2}}{2}} \end{array}$

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