GATE ME 2018 Set 1 — Question 8

Let $L_t$ be the length of the train and $L_p$ be the length of the platform, both in meters.
Let $v_t$ be the speed of the train and $v_m$ be the speed of the man.First, convert the speeds from km/h to m/s:
Speed of the train, $v_t = 54 \text{ km/h} = 54 \times \frac{1000 \text{ m}}{3600 \text{ s}} = 54 \times \frac{5}{18} \text{ m/s} = 15 \text{ m/s}$.
Speed of the man, $v_m = 9 \text{ km/h} = 9 \times \frac{5}{18} \text{ m/s} = 2.5 \text{ m/s}$.Case 1: Train crossing the platform
When the train crosses the platform, the total distance covered by the front of the train is the sum of the length of the platform and the length of the train.
Distance = $L_p + L_t$
Time taken, $t_1 = 25$ s.
Speed = $v_t = 15$ m/s.
Using the formula, Distance = Speed × Time:
$L_p + L_t = v_t \times t_1$
$L_p + L_t = 15 \times 25$
$L_p + L_t = 375$ (Equation 1)Case 2: Train passing the man
Since the man is running in the same direction as the train, we need to use the relative speed of the train with respect to the man.
Relative speed, $v_{rel} = v_t - v_m = 15 - 2.5 = 12.5$ m/s.
The distance covered by the train to pass the man is its own length, $L_t$.
Time taken, $t_2 = 14$ s.
Using the formula, Distance = Speed × Time:
$L_t = v_{rel} \times t_2$
$L_t = 12.5 \times 14$
$L_t = 175$ meters.Now, substitute the value of $L_t$ into Equation 1 to find the length of the platform, $L_p$:
$L_p + 175 = 375$
$L_p = 375 - 175$
$L_p = 200$ meters.So, the length of the train is 175 m and the length of the platform is 200 m.
This corresponds to option (D).