Two Trains and a Platform
Two trains run on parallel straight tracks. Train A is 180 m long and travels at 20 m/s. Train B is 120 m long and travels at 30 m/s in the same direction. A platform is 200 m long. A passenger on the platform walks at 1.5 m/s in the same direction as the trains. The controller must decide overtaking times and whether a signal placed at the far end of the platform is passed safely.
1. Find the time from the moment the front of B reaches the rear of A until B has completely passed A.
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Overtaking requires the front of B to close the gap and then B's rear to clear A's front. The relative speed is 30 − 20 = 10 m/s. The distance the front of B must gain relative to A is the sum of lengths: 180 + 120 = 300 m. Time = 300/10 = 30 s. Judgement point: students often use only one train's length or the platform length; the platform is irrelevant to the overtaking calculation because both trains are on parallel tracks and their relative motion does not involve the platform. The answer is 30 s.
2. A signal is at the far end of the platform. How long does train A take to pass the signal, and does the walking passenger change this?
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For train A to pass a fixed signal, its front must travel from the signal to its rear clearing it, a distance equal to its own length, 180 m, at 20 m/s: t = 180/20 = 9 s. The platform length (200 m) does not enter this calculation - the signal is a point. The walking passenger does not change when the train passes the signal; the passenger's motion only matters if we ask when the passenger and a particular point on the train are level. The judgement is to separate 'train passing a fixed point' (use train length) from 'train passing a platform' (use train length + platform length).
3. How long is any part of train B alongside the walking passenger if they move in the same direction?
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Relative speed of B with respect to the passenger = 30 − 1.5 = 28.5 m/s. The length that must sweep past the passenger is B's full length, 120 m. Time = 120/28.5 ≈ 4.21 s. If the passenger walked the opposite way, the relative speed would be 31.5 m/s and the time ≈ 3.81 s. The key judgement is that 'alongside' means the whole train passes the passenger, so the train's length is the relevant distance, and the passenger's speed must be combined with the train's speed as a relative velocity, not added to the train's length.