The speed of a train T is 100 km per hour and the speed of a person P is 4 km per hour. T crosses P in 15 seconds, if P travels along the direction of motion of T. If P travels along the opposite direction of T, then in how much time does T cross P, in seconds, approximately?

Correct Option: C (13.85)

The question evaluates foundational principles of one-dimensional kinematics and relative speed, which form a core component of the CSAT aptitude syllabus.

Step 1: Determine the relative speed in the same direction. When two bodies move in the same direction, their relative speed is the difference between their individual speeds. Relative Speed (v_rel1) = Speed of Train (T) - Speed of Person (P) = 100 km/hr - 4 km/hr = 96 km/hr. To standardise units, we must convert this to metres per second (m/s) by multiplying by 5/18: 96 × 5/18 = 80/3 m/s.

Step 2: Calculate the length of the train. The distance the train must cover to cross the person is strictly equal to the length of the train (L). L = v_rel1 × Time = 80/3 m/s × 15 s = 400 meters.

Step 3: Determine the time taken to cross in opposite directions. When bodies move in opposite directions, their relative speed is the sum of their individual speeds. Relative Speed (v_rel2) = 100 km/hr + 4 km/hr = 104 km/hr. Convert to m/s: 104 × 5/18 = 260/9 m/s. Time taken to cross (t) = Distance/Speed = 400/(260/9) = 3600/260 = 180/13 seconds. t ≈ 13.846 seconds, which formally rounds to 13.85 seconds (Option C).

Analysis of Incorrect Options:

• Option A (13.51): Incorrect. This value typically arises from arithmetic errors during unit conversions, leading to an artificially lower time quotient.
• Option B (13.65): Incorrect. This is mathematically invalid and likely derived from miscalculating the fractional division of 3600 ÷ 260.
• Option D (14.05): Incorrect. This represents an overestimation resulting from failing to add the speeds together (perhaps using the initial 96 km/hr relative speed incorrectly).

Concluding Takeaway: To master relative motion problems efficiently, remember the core Galilean framework: always subtract speeds for parallel movement (same direction), add speeds for opposing movement (opposite direction), and ensure strict unit parity before calculating.