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## PROBLEMS ON TRAINS -> IMPORTANT FORMULAE

 1. a km/hr = [a * 5/18]m/s. 2. a m/s = [a * 18/5] km/hr. 3. Time taken by a trian of length l metres to pass a pole or a standing man or a signal post is equal to the time taken by the train to cover l metres. 4. Time taken by a train of length l metres to pass a stationary object of length b metres is the time taken by the train to cover (l + b) metres. 5. Suppose two trains or two bodies are moving in the same direction at u m/s and v m/s, where u>v, then their relatives speed = (u - v) m/s. 6. Suppose two trains or two bodies are moving in opposite directions at u m/s and v m/s, then their relative speed is = (u + v) m/s 7. If two trains of length a metres and b metres are moving in opposite directions at u 8. If two trains of length a metres and b metres are moving in the same direciton at u m/s and v m/s, then the time taken by the faster train to cross the slower train = (a + b)/(u - v) sec. 9. If tow trains (or bodies) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then (A’s speed) : (B’s speed) = (√b : √a).

## PROBLEMS ON TRAINS -> SOLVED EXAMPLES

1. Two tain 100 metres and 120 metres long are running in the same direction with speeds of 72 km/hr and 54 km/hr. In how much time will the first train cross the second? Sol. Relative speed of the train = (72 - 54) km/hr = 18 km/hr = [18 * 5/18] m/sec = 5 m/sec. Time taken by the trains to cross each other = Time taken to cover (100 + 120) m at 5 m/sec = [220/5]sec = 44 sec. A train 220 m long is running with a speed of 59 kmph. In what time will it pas a man who is running at 7 kmph in the direction opposite to that in which the tain is going? Sol. Speed of the train relative to man = (59 + 7) kmph = [66 * 5/18] m/sec = [55/3] m/sec. Time taken by the train to cross the man = Time taken by it to cover 220m at [55/3] m/sec = [220 * 3/55] sec = 12 sec. A man siting in a trian which is travelling at 50 kmph observes that a goods trian, travelling in opposite direction, takes 9 seconds to pass him. If the goods train is 280 m long, find its speed. Sol. Relative speed = [280/9] m/sec = [280/9 * 18/5] kmph = 112 kmph. ∴ Speed of goods train = (112 - 50) kmph = 62 kmph.

## PROBLEMS ON TRAINS -> Exercise

1. A man is standing on a railway bridge which is 180 m long. He finds that a train crosses the bridge in 20 seconds but himself in 8 seconds. Find the length of the train and its speed.

• A. 36
• B. 54
• C. 67
• D. 76
 Ans: B. Sol. Let the length of the train be x metres. Then, the train covers x metres in 8 seconds and (x + 180) metres in 20 seconds. ∴ x/8 = x + 180 / 20 ⇔ 20x = 8(x + 180) ⇔ x = 120. ∴ Length of the train = 120 m. Speed of the train = [120/8]m/sec = m/sec = [15 * 18/5]kmph = 54 kmph.

2. Two trains are running at 40 km/hr and 20 km/hr respectively in teh same direction. Fast train completely passes a man sitting in the slower train in 5 seconds. What is the length of the fast train?

• A. 27 7/9 m
• B. 28 m
• C. 29
• D. 30
 Ans: A. Sol. Relative speed = (40-20) km/hr = [20 * 5/18] m/sec = [50/9] m/sec. Length of faster train = [50/9 * 5] m = 250/9 m = 27 7/9 m.

3. Two train travel in opposite directions at 36 kmph and 45 kmph and a aman sitting in slower train passes the faster train in 8 seconds. Then length of the faster train is:

• A. 120
• B. 140
• C. 180
• D. 190
 Ans: C. Sol. Relative speed = (36 + 45) km/hr = [81 * 5/18] m/sec = [45/2] m/sec. Length of train = [45/2 * 8] m = 180 m.