Problems on Trains

A train crossing something covers its own length plus that thing's length. A pole or a man counts as length zero.

Start with this

A train is not a dot. To pass a pole it must move its whole length. To cross a platform it must move its length plus the platform's.

Train questions are time and distance with one twist: the train's own length is part of the distance.

Pole or person vs platform or bridge

Worked example

A train 240 m long passes a pole in 12 seconds. How long will it take to cross a platform 360 m long?

1. Speed = 240 / 12 = 20 m/s.
   Why: Passing a pole: distance = train length.
2. Platform: distance = 240 + 360 = 600 m.
3. 600 / 20 = 30 seconds.

Answer: 30

Two trains

Two trains passing each other cover the SUM of their lengths. Opposite directions: relative speed = sum of speeds. Same direction: difference.

Worked example

Two trains 150 m and 100 m long run on parallel tracks at 54 km/h and 36 km/h in opposite directions. How long to cross each other?

1. Relative speed = 54 + 36 = 90 km/h = 25 m/s.
   Why: Opposite directions add; convert to m/s.
2. Distance = 150 + 100 = 250 m.
   Why: Both lengths.
3. 250 / 25 = 10 seconds.

Answer: 10

Worked example

A 200 m train at 72 km/h overtakes a man walking at 6 km/h in the same direction. How long does it take?

1. Relative speed = 72 - 6 = 66 km/h = 55/3 m/s.
   Why: Same direction subtracts.
2. 200 / (55/3) = 600/55 = 120/11 seconds, about 10.9 s.

Answer: 120/11 s (about 10.9 s)

Shortcut: Pole and platform together

Use it when: 'Crosses a pole in 15 s and a 100 m platform in 25 s'.

1. The extra 10 s is the platform: speed = 100/10 = 10 m/s.
2. Length = 10 × 15 = 150 m.

Pole and platform together: try it

As above. Train length?

1. 150 m.

Answer: 150 m

Your turn medium

Trains 120 m and 180 m long run towards each other at 50 and 40 km/h. Time to cross?

1. 90 km/h = 25 m/s. 300/25.

Answer: 12 s

Your turn medium

Trains 100 m and 80 m, same direction, 60 and 42 km/h. Time for the faster to pass the slower?

1. 18 km/h = 5 m/s. 180/5.

Answer: 36 s

Your turn hard

Two trains leave A and B towards each other at the same time. After meeting, they take 9 h and 16 h to reach B and A. Ratio of speeds?

1. √(16) : √(9).

Answer: 4:3

Your turn hard

A train at 54 km/h passes a man walking at 6 km/h in the same direction in 12 s. Length of the train?

1. Relative 48 km/h = 40/3 m/s.
2. 40/3 × 12 = 160 m.

Answer: 160 m

Recap

• add the platform length when crossing it
• add both lengths when two trains pass
• use relative speed for trains and walkers

Now practise

9 questions with full solutions.

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