Question

An engine of a train, moving with uniform acceleration, passes the signal-post with velocity $u$ and the last compartment with velocity $v$. The velocity with which middle point of the train passes the signal post is :

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

Hint:

For any object moving with constant acceleration, the velocity at the geometric midpoint of a distance interval is the {Root Mean Square} of the initial and final velocities.

Answer 1

The Correct Option is C

Step 1: Let the length of the train be $L$ and acceleration be $a$.
Step 2: Using 3rd equation of motion for the whole train: $v^2 = u^2 + 2aL \Rightarrow 2aL = v^2 - u^2$.
Step 3: Let $v_m$ be the velocity at the middle point ($x = L/2$).
Step 4: $v_m^2 = u^2 + 2a(L/2) = u^2 + aL$.
Step 5: Substitute $aL = \frac{v^2 - u^2}{2}$: $v_m^2 = u^2 + \frac{v^2 - u^2}{2} = \frac{2u^2 + v^2 - u^2}{2} = \frac{u^2 + v^2}{2}$.
Step 6: $v_m = \sqrt{\frac{u^2 + v^2}{2}}$.