JEE PYQ: Motion in a Straight Line - Question ID 182ce3644b8f (JEE Main 2023)

Core Formula & Concept:

When two objects move in opposite directions, their relative velocity is the sum of their individual velocities. This is a direct consequence of the principle of relative motion in one dimension.

Key formulas:

• Relative velocity when moving in opposite directions: $v_{\text{rel}} = v_A + v_B$ where $v_A$ and $v_B$ are the speeds of the two objects.
• Distance covered under constant relative velocity: $\text{Distance} = v_{\text{rel}} \times \text{Time}$ This distance corresponds to the length of the object being observed (here, train B).

Step-by-Step Derivation:

Step 1: Convert velocities to consistent units (m/s)

Train A moves at $90 \text{ km/h}$. Convert to m/s: $90 \times \frac{1000 \text{ m}}{3600 \text{ s}} = 25 \text{ m/s}$

Train B moves at $54 \text{ km/h}$. Convert to m/s: $54 \times \frac{1000 \text{ m}}{3600 \text{ s}} = 15 \text{ m/s}$

Step 2: Compute relative velocity

Since the trains move in opposite directions, their relative velocity is the sum of their speeds: $v_{\text{rel}} = v_A + v_B = 25 + 15 = 40 \text{ m/s}$

Step 3: Calculate the length of train B

The passenger in train A observes train B for $8 \text{ s}$. During this time, the entire length of train B passes by the passenger. Hence, the length of train B is: $\text{Length of train B} = v_{\text{rel}} \times \text{Time} = 40 \times 8 = 320 \text{ m}$

Step 4: Match with given options

The calculated length is $320 \text{ m}$, which corresponds to option D.

Common Traps & Exam Tip:

Trap 1: Incorrect relative velocity
Students often subtract velocities when objects move in opposite directions, leading to $v_{\text{rel}} = 25 - 15 = 10 \text{ m/s}$ and an incorrect length of $80 \text{ m}$ (option A). Always remember: opposite directions → add velocities.

Trap 2: Unit inconsistency
Failing to convert km/h to m/s results in nonsensical answers. Always ensure consistent units (preferably SI units) before calculations.

Exam Tip:
Visualize the scenario: two trains moving toward each other. The faster they approach, the quicker the second train passes by. This intuition reinforces that relative velocity is the sum of their speeds.