How to Solve Man vs Train Crossing Problems
Solve train-crossing-a-person aptitude problems using relative speed for same and opposite directions, with worked examples.
Expected Interview Answer
When a train crosses a person walking or running, the train covers its own length L relative to that person at their relative speed, so time = L / (relative speed), where relative speed is the sum of both speeds if moving toward each other and the difference if moving in the same direction.
The distance that must be covered when a train crosses a moving person is just the train’s length L, because the person is treated as a point compared to the train — but the speed used is the relative speed between them, not the train’s speed alone. If the train and the person move toward each other, their closing speed is the sum of both speeds, so the crossing happens quickly: time = L/(trainSpeed+personSpeed). If they move in the same direction, the train only gains on the person at the speed difference, so time = L/(trainSpeed−personSpeed), which is longer. This is a special case of the general two-body relative-speed framework, with the person’s “length” treated as zero and the train contributing the only physical length.
- Treating the person as a point (length zero) simplifies the setup to just the train’s length
- Same toward/away relative-speed rules apply as in races and trains-crossing-trains problems
- One formula, time = L/(relative speed), covers both direction cases
AI Mentor Explanation
Think of a sightscreen being wheeled past a fielder standing near the boundary — the sightscreen’s own width is the only physical length involved (the fielder is treated as a point), and how fast it “crosses” the fielder depends on their relative speed: if the fielder is walking toward the moving sightscreen, the crossing happens fast (speeds add); if walking the same way, it takes longer (speeds subtract). This mirrors exactly how a train’s length crosses a walking or running person at their relative speed.
Worked example
Train speed
- 54 km/h = 15 m/s
Relative speed (same direction)
- 15 − 5/3 = 40/3 m/s
Crossing time
- 150 ÷ (40/3) = 11.25 s
Step-by-Step Explanation
Step 1
Convert all speeds to the same unit
Convert km/h to m/s (multiply by 5/18) if the train length is in meters.
Step 2
Determine relative speed
Opposite directions: add speeds. Same direction: subtract speeds.
Step 3
Identify the distance to cross
The distance is just the train’s length L — the person is treated as a point.
Step 4
Apply time = distance / relative speed
Divide the train’s length by the relative speed computed above.
What Interviewer Expects
- Correct unit conversion between km/h and m/s
- Correct relative-speed setup for same vs opposite direction
- Recognizing the person contributes no length, only the train does
- Correct final division of length by relative speed
Common Mistakes
- Using the train’s speed alone instead of the relative speed to the person
- Adding speeds when the motion is in the same direction (should subtract)
- Forgetting to convert km/h to m/s before dividing by a length in meters
- Mistakenly adding the person’s own length as if they were a train
Best Answer (HR Friendly)
“I treat the person as having no length of their own, so the only distance the train has to cover is its own length. What matters is the relative speed between the train and the person — if they are moving toward each other, I add the speeds, and if they are moving the same way, I subtract them, since the train is really just gaining on the person at that difference. Then it is simply the train’s length divided by that relative speed, after making sure both speeds are in the same units.”
Follow-up Questions
- How does this change when a train crosses another train instead of a person?
- How would you find the length of the train if only the crossing time and speeds are given?
- How does crossing a stationary platform differ from crossing a moving person?
- How would wind or an escalator-style moving platform affect this setup?
MCQ Practice
1. A 120m train moving at 60 km/h crosses a man standing still. How long does it take?
60 km/h = 50/3 m/s. Time = 120 ÷ (50/3) = 7.2s.
2. A 200m train at 72 km/h (20 m/s) crosses a man running at 8 km/h in the OPPOSITE direction. Relative speed is?
8 km/h = 20/9 m/s ≈ 2.2 m/s. Opposite direction: 20 + 2.2 ≈ 22.2 m/s.
3. When a train crosses a man walking in the SAME direction as the train, the relative speed used is?
Same-direction motion uses the difference of speeds, since the train only gains on the man at that rate.
Flash Cards
Distance to cross when a train crosses a person? — Just the train’s length L — the person is treated as a point.
Relative speed, opposite directions? — Sum of the train’s speed and the person’s speed.
Relative speed, same direction? — Difference between the train’s speed and the person’s speed.
Core crossing-time formula? — Time = Train length / Relative speed.