JEE PYQ: Motion in a Straight Line - Question ID 6bd5e2b4601d (JEE Main 2023)

Core Formula & Concept:

In problems involving trains crossing tunnels, the effective distance a train must cover is the sum of the tunnel length ($L$) and the train’s own length ($l$ or $4l$). This is because the train is considered to have completely crossed the tunnel only when its last carriage exits the tunnel.

The key formula is: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$ where the distance is $L + \text{length of the train}$, and the speed must be in consistent units (here, converted from km/h to m/s).

Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds. However, in this problem, each train’s time to cross the tunnel is calculated independently, so relative speed is not directly needed.

Step-by-Step Derivation:

Step 1: Convert velocities to m/s

Train A’s speed: $108 \text{ km/h} = 108 \times \frac{1000}{3600} = 30 \text{ m/s}$
Train B’s speed: $72 \text{ km/h} = 72 \times \frac{1000}{3600} = 20 \text{ m/s}$

Step 2: Express the time taken by each train to cross the tunnel

For Train A (length $l$): $t_A = \frac{L + l}{30}$
For Train B (length $4l$): $t_B = \frac{L + 4l}{20}$

Step 3: Use the given time difference

The problem states that Train A takes $35 \text{ s}$ less time than Train B: $t_B - t_A = 35$ Substitute the expressions for $t_A$ and $t_B$: $\frac{L + 4l}{20} - \frac{L + l}{30} = 35$

Step 4: Solve for $L$ in terms of $l$

Multiply through by $60$ (LCM of 20 and 30) to eliminate denominators: $3(L + 4l) - 2(L + l) = 35 \times 60$ Simplify: $$3L + 12l - 2L - 2l = 2100 \\ L + 10l = 2100$$ Given that $L = 60l$ (from the problem statement), substitute: $$60l + 10l = 2100 \\ 70l = 2100 \\ l = 30 \text{ m}$$ Now, compute $L$: $L = 60l = 60 \times 30 = 1800 \text{ m}$

Step 5: Verify the answer matches the options

The calculated tunnel length is $1800 \text{ m}$, which corresponds to option C.

Common Traps & Exam Tip:

Trap 1: Forgetting to add the train’s length to the tunnel length. Students often mistakenly use only $L$ as the distance, ignoring the train’s own length.

Trap 2: Incorrect unit conversion. Failing to convert km/h to m/s leads to wrong numerical results. Always double-check unit consistency.

Trap 3: Misinterpreting the time difference. Some students reverse the order of $t_A$ and $t_B$, leading to a negative time difference. Carefully read the problem statement to avoid this.

Exam Tip: When dealing with trains and tunnels, always draw a quick sketch to visualize the scenario. This helps in identifying the correct effective distance (tunnel length + train length).