Train Crossing Problems: Which Length Must Clear?

Track the front and the rear

A train is not a point. “Completely crosses” means its rear has also cleared the person, pole or platform. Draw the first and last positions before choosing the distance. Assume a straight track, constant speeds and negligible person/pole width.

Passing a stationary pole

A 120 m train travels at 54 km/h. First convert the speed: 54 × 5/18 = 15 m/s. Start timing when its front reaches the pole. Stop when its rear reaches the same pole. The front has moved one train length, 120 m.

Time = length / speed = 120/15 = 8 seconds. Check: 15 m every second for 8 seconds gives 120 m. The distance is not zero merely because the pole is narrow.

Passing a platform

The same 120 m train passes a 180 m platform at 15 m/s. At the start, the train's front is at the platform entrance. To finish, its rear must pass the far end. The front first moves 180 m to the far end, then another 120 m to pull the rear clear.

Required travel = platform length + train length = 180 + 120 = 300 m. Time = 300/15 = 20 seconds. This addition comes from the two positions; it is not a formula to apply to every question.

Passing a moving person

The 120 m train travels at 54 km/h and overtakes a person moving at 18 km/h in the same direction. Relative speed = 54 − 18 = 36 km/h = 36 × 5/18 = 10 m/s. In the person's viewpoint, the full 120 m train must pass.

Time = 120/10 = 12 seconds. If the person instead moves towards the train at 18 km/h, relative speed is 54 + 18 = 72 km/h = 20 m/s, so the crossing takes 6 seconds. Direction changes the relative speed, not the train's length.

A common mistake

Using platform length alone stops the clock when the train's front reaches the far end, while much of the train is still beside the platform. Likewise, 120/54 mixes metres and km/h. State what starts and stops the timing and make units consistent.

Check yourself

1. A 150 m train at 10 m/s crosses a pole. Time?
2. The same train crosses a 250 m platform at 10 m/s. Time?
3. A 180 m train overtakes a person at relative speed 9 m/s. Time?

Answers: (1) 150/10 = 15 s. (2) (150 + 250)/10 = 40 s. (3) 180/9 = 20 s.

These are original explanatory examples, not attributed to an exam paper.