JEE PYQ: Motion in a Straight Line - Question ID 63a001282c8f (JEE Main 2021)

In problems involving uniformly accelerated motion in a straight line, the following kinematic relations are fundamental:

• Velocity–displacement relation (no time involved): $v^2 - u^2 = 2\,a\,s$ where
  $u$ = initial velocity,
  $v$ = final velocity,
  $a$ = uniform acceleration,
  $s$ = displacement.
• Velocity at any intermediate point can be found by applying the same relation to the segment of interest.

Here the train’s engine (front) passes the signal post with speed $u$ and the last compartment (rear) passes it with speed $v$. We are to find the speed of the middle point of the train as it passes the same signal post.

1. Let the total length of the train be $L$. 2. When the engine (front) passes the signal post, its velocity is $u$. 3. When the last compartment (rear) passes the same post, the engine has moved forward by distance $L$ and its velocity has increased to $v$. 4. Apply the velocity–displacement relation over the entire train length $L$: $v^2 - u^2 = 2\,a\,L \quad\Longrightarrow\quad a\,L = \frac{v^2 - u^2}{2}.$ 5. Now consider the middle point of the train. As the engine moves from the signal post to the position where the middle point reaches the post, the engine covers a distance $\tfrac{L}{2}$. 6. Let $w$ be the velocity of the middle point when it passes the post. Applying the same kinematic relation over the half-length $\tfrac{L}{2}$ gives $w^2 - u^2 = 2\,a\,\Bigl(\tfrac{L}{2}\Bigr) = a\,L.$ 7. Substitute $a\,L$ from step 4: $w^2 - u^2 = \frac{v^2 - u^2}{2} \quad\Longrightarrow\quad w^2 = u^2 + \frac{v^2 - u^2}{2} = \frac{v^2 + u^2}{2}.$ 8. Therefore the required velocity is $w = \sqrt{\frac{v^2 + u^2}{2}}.$

• Many students mistakenly take the arithmetic mean $\tfrac{u+v}{2}$ (option A), forgetting that velocity under uniform acceleration does not vary linearly with distance.
• Others confuse the velocity–displacement relation with the velocity–time relation and arrive at incorrect expressions.
• Exam Tip: Always write down the kinematic relation that connects the known quantities (here $v^2 - u^2 = 2\,a\,s$) and then apply it to the segment of interest.