# Problems on Trains - Time, Speed & Distance

• Authored by Nitin Gupta, Founder, Director at AlphaNumeric.

Train crossing a pole/man/platform/bridge/train
Whenever one object of finite length crosses another object of finite length, in crossing the object completely it covers a distance equal to the sum of the lengths. The distance to be covered is independent of the direction in which it is crossing i.e. whether the two objects are crossing each other in opposite directions or if one is overtaking the other i.e. in same direction, the distance to be covered for one to completely cross the other is the sum of the lengths.

In the problems asked, usually a train is crossing either a pole or a man (stationary or walking) or a platform or bridge or another train (obviously moving). For all such cases you could use the following with the modifications as noted below:
Time taken to cross = (L1 + L2)/(S1 +- S2)

If one of the object is a pole or a man, its length will be 0 (zero)
If one of the object is stationary (e.g. pole, platform, bridge), its speed will be 0 (zero) and the relative speed will just be the speed of the moving object

A train running at 72 kmph crosses a telephone pole in 7 sec. What is the length of the train?

Since a telephone pole is a stationary object of negligible length, the distance the train covers is just its own length.
Converting 72 kmph into m/s, since 72 is 4th multiple of 18,
speed in m/s is 4th multiple of 5 i.e. 20 m/s
Distance = 20 × 7 = 140 m

A train crosses 2 platforms of length 400 m and 600 m in 6 seconds and 8 seconds respectively. What is the length of the train?

Approach 1:
(400 + L)/6 = (600 + L)/8 => L = 200

Approach 2: Using proportionality
In the two scenarios ‘crossing platform of 400 m in 6 sec’ and ‘crossing platform of 600 m in 8 sec’, the speed is the same. Hence distance will be proportional to time taken. The distances covered are 400 + x and 600 + x i.e. a difference of 200. And the time taken are 6 sec & 8 sec. Thus ratio of time is 3 : 4. This will also be the ratio of distances and we know that the difference in distances is 200. Thus, the distances covered in the two
scenarios is 600 m and 800 m. Thus, length of train is 200 m

A train crosses two persons who are walking at 2 kmph and 4 kmph, in the same direction in which the train is going, in 9 and 10 seconds respectively. Find the length of the train.

In the two scenarios, ‘train crossing man walking at 2 kmph’ and ‘train crossing man walking at 4 kmph’, the distance covered will be the length of the train itself i.e. will be same in both the cases. Thus, ratio of speed will be inverse of ratio of time taken. Since ratio of time taken is 9 : 10, ratio of speed will be 10 : 9. Further we also know that the difference in the speeds in the two scenario will be 2 kmph (speeds will be S – 2 and S – 4). Thus, the speeds in the two scenario are 20 kmph and 18 kmph. At 18 kmph i.e. 5 m/s, distance covered in 10 sec will be 50 m. This has to be the length of the train.

A tunnel measuring 4 km and 636 meters is designed specifically for two trains to pass simultaneously in the same or opposite directions. Therefore two express trains of length 400 m each, travel through the
tunnel at the rate 56 kmph and 80 kmph.
Q1) Assuming that both the trains enter the tunnel at the same point of time (t = 0) from the two different ends, then the minimum value of ‘t’ such that both the trains have cleared the tunnel will be.
a. 2 min 45 sec
b. 2 min 24 sec
c. 2 min 36 sec
d. None of these
Q2) Assuming that the guard starts walking immediately as he enters the tunnel and the train is traveling at the speed of 80 kmph, what is the time by taken a guard walking at the rate of 5 kmph along the
corridor towards the engine to clear the tunnel?
a. 3 min 46 sec
b. 3 min 53 sec
c. 4 min
d. None of these

Q1) Distance to be covered 4636 + 400 = 5036 m
Speed of the slower train = 56 kmph = 56 × (5 / 18) m
∴ sec
∴ time taken = 5.39 min.
Q2) Time taken = 4636/(80+5)*5/18 = 196 sec. = 3 min, 16 sec

Practice Questions

1. Two trains are travelling in opposite direction at uniform speed 60 and 50 km per hour respectively. They take 5 seconds to cross each other. If the two trains had travelled in the same direction, then a passenger sitting in the faster moving train would have overtaken the other train in 18 seconds. What are the lengths of trains (in metres)?
a) 112.78
b) 97.78, 55
c) 102.78, 50
d) 102.78, 55

2. Laxman and Bharat decide to go from Agra to Delhi for watching a cricket match and board two different trains for that purpose. While Laxman takes the first train the leaves for Delhi, Bharat decides to wait for some time and take a faster train. On the way, Laxman sitting by the windowseat noticed that the train boarded by Bharat crossed him in 12 seconds. Now the faster train can travel 180 km in three hours, while the slower train takes twice as much time to do it. Given this, mark all the correct options.
a. If the faster train has taken 30 seconds to cross the entire length of the slower train, the difference between the lengths of the two trains is 50 m.
b. If the faster train had been running twice as much faster, it would have taken 10 seconds to overtake the slower train.
c. Had the faster train taken 24 seconds to cross the entire length of the slower train, the length of the slower train would have been 100 m.
d. If the slower train had been running at one and a half times of its current speed, the faster train would have taken 24 seconds to overtake Laxman. (IIFT 2006)