Train.............[competitive. exam prep.... topic 28]

Trains Shortcut Methods :
Problems on trains are most frequently asked questions in any competitive exam.
Problems on trains and ‘Time and Distance’ are almost same. The only difference is we have to consider the length of the train while solving problems on trains.

Points To Remember
  • Time taken by a train of length of L meters to pass a stationary pole is equal to the time taken by train to cover L meters.
  • Time taken by a train of length of L meters to pass a stationary object of length P meters is equal to the time taken by train to cover (L + P) meters.
  • If two trains are moving in same direction and their speeds are x km/h and y km/h (x > y) then their relative speed is (x – y) km/h.
  • If two trains are moving in opposite direction and their speeds are x km/h and y km/h then their relative speed is (x + y) km/h.

Unit Conversion
  • km/h to m/sec conversion
  • m/sec to km/h unit conversion
Some Shortcut Methods
Rule 1:
If two trains of p meters and q meters are moving in same direction at the speed of x m/s and y m/s (x > y) respectively then time taken by the faster train to overtake slower train is given by
Example
Two trains of length 130 meter and 70 meter are running in the same direction with the speed of 50 km/h and 70 km/h. How much time will faster train take to overtake the slower train from the moment they meet?
Sol:
Let’s say p = 130 meter = 0. 13 km
q = 70 meter = 0.07 km
x = 70 km/h and y = 50 km/h,
So from the equation given above,
0.01 hours = 36 second
So it will take 36 seconds to overtake.

Rule 2:
If two trains of p meters and q meters are moving in opposite direction at the speed of x m/s and y m/s respectively then time taken by trains to cross each other is given by
Example
Two trains of length 130 meter and 70 meter are running in the opposite direction with the speed of 50 km/h and 70 km/h. How much time will trains take to cross each other from the moment they meet?
Sol:
Let’s say p = 130 meter = 0. 13 km
q = 70 meter = 0.07 km
x = 70 km/h and y = 50 km/h,
So from the equation given above,
0.0017 hours = 6 seconds
So it will take 6 seconds to cross each other.

 Another shortcut methods :


1. When a train passes a stationary point, the distance covered (in the passing) is the length of the train.
Example 1: What is the time taken by a train of length 360m to cross a pole at a speed of 72 km/h ?
Ans: Time taken by the train to cross the pole = Length of the train / Speed of the train
Speed of the train is given in km/h, whereas the length of the train is given in mts. So the speed of the train is to be expressed in m/sec.
Speed of the train ( in m/sec) = 72 x (5/18)
= 20 m/sec
Time taken by the train to cross the pole = 360/20
= 18 seconds.

2. If the train is crossing a platform or a bridge, the distance covered by the train is equal to the length of the train plus the length of the platform or a bridge.
Example 2: How long will a train 200 m long traveling at a speed of 54 km/h take to cross a platform of length 100 m?
Ans: Distance covered by the train = Length of the train + Length of the platform
= 200 + 100
= 300 m
Speed of the train is given in km/h, whereas the distance covered by the train is given in mts. So speed of the train is to be expressed in m / sec.
Speed of the train = 54 x (5/18)
= 15 m/ sec
Time taken by the train = 300/15
= 20 seconds.

3. If two trains pass each other ( traveling in the same direction or in opposite directions) , the total distance covered ( in the crossing/ overtaking as the case may be) is equal to the sum of the lengths of the two trains.
Example 3: Two trains 121 m and 99 m in length respectively are running in opposite directions, one at the rate of 40 km/h and the other at the rate of 32 km/h. In what time will they be completely clear of each other from the moment they meet?
Ans: As the trains are moving in opposite directions their relative speed = 40 + 32 km/hr.
= 72 km/hr.
The length to be traveled by the trains = 121 + 99
= 220 m.
The speed of the train is given in Km/h, whereas the length is given in m. Hence, the relative speed of the trains is to be expressed in m/sec.
The relative speed in m/sec = 72 x (5/18)
= 20 m /sec
Time required to completely clear of each other from the moment they meet = 220 / 20
= 11 secs.

5. If two bodies are moving in opposite direction at speeds S1 and S2 respectively, then the relative speed is:Relative speed = S1+ S2.

6. Two trains of length ‘p’ m and ‘q’ m respectively run on parallel lines of rails. When running in the same direction the faster train passes the slower one in ‘a’ seconds, but when they are running in opposite directions with the same speeds as earlier, they pass each other in ‘b’ seconds. Then,
Speed of the faster train = [( p + q)/ 2] x [ ( a+b) / (a x b)]
Speed of the slower train = [(p-q) / 2] x [ (a-b) / (a x b)]

Note : The speeds obtained using the above formula are in m/ sec, if the speeds are to be expressed in km/h, they have to be multiplied by 18/5.

Example 4: Two trains of length 100 m and 250 m run on parallel lines. When they run in the same direction it will take 70 seconds to cross each other and when they run in opposite direction, they take 10 seconds to cross each other. Find the speeds of the two trains?
Ans: Speed of the faster train = [(100 + 250) / 2] [ (70 + 10) / ( 70 x 10) ].
= 175 x (8 /70)
= 20 m/sec.
Speed of the slower train = [ ( 100 + 250) / 2] [ ( 70-10) / (70 x 10) ]
= 175 x ( 6/ 70)
= 15 m/ sec.
Therefore, speeds of the trains are 72 km/h (20 x 18 /5) and 54 km/h ( 15 x 18/5) respectively.

7. If a train passes by a stationary man in ‘p’ seconds and passes by a platform / bridge, the length of which is ‘m’ m, completely in ‘q’ sec. Then
Length of the train = (m x p) / (q-p).

Example  5: A train crosses by a stationary man standing on the platform in 7 seconds and passes by the platform completely in 28 seconds. If the length of the platform is 330 meters, what is the length of the train?
Ans: Length of the train = ( 330 x 7) / ( 28-7)
= 330x 7 / 21
= 110 m.

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