Concept of Trains

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Basic Concept with Examples

FORMULAS:

Speed of the train = Distance Travelled / Time Taken
Units of speed are m/sec or km/hr
A km/hr = (5A/18) m/sec
A m/sec = (18A/5) km/hr

1) Time taken by a train of length L meters, travelling at a speed of S m/sec to cross a pole/a standing man/a signal post/any other object(of negligible length) is L/S sec

Example: A train covers 85 m in passing a signal post. What is the length of the train?
Length of the train = 85 m

2) Time taken by a train of length L meters, travelling at a speed of S m/sec to cross/pass a stationary object (platform/a standing train/a bridge) of length P meters is (L+P)/S sec

Example: A 28 m long train passes a platform which is 85 m long. Find the distance covered by the train in passing the platform.
Required distance = Length of train + Length of platform = 28 + 85 = 113 m

3) If two trains are moving in opposite directions, then their relative speed is equal to the sum of the speeds of both the trains.

Example: Two trains are moving in opposite directions with speeds of 4 m/sec and 7 m/sec, respectively. Find their relative speed.
Required relative speed = 4 m/sec + 7 m/sec = 11 m/sec

4) If two trains are moving in same directions, then their relative speed is equal to the difference of the speeds of both the trains.

Example: Two trains are moving in same directions with speeds of 19 km/hr and 25 km/hr, respectively. What will be the relative speed of the train running at 25 km/hr in respect of the train running at 19 km/hr?
Required relative speed = 25 km/hr – 19 km/hr = 6 km/hr

5) Time taken by a train of length L meters, travelling at a speed of S1 m/sec to cross a man travelling at a speed S2 m/sec in the same direction is L/(S1 - S2 ) sec

6) Time taken by a train of length L meters, travelling at a speed of S1 m/sec to cross a man travelling at a speed S2 m/sec in the opposite direction is L/(S1 + S2 ) sec

7) Time taken by a train of length L1 meters, travelling at a speed of S1 m/sec to cross a man sitting in another train of length L2 meters, travelling at a speed of S2 m/sec in the same direction is L1/(S1 - S2) sec where S1 > S2

8) Time taken by a train of length L1 meters, travelling at a speed of S1 m/sec to cross a man sitting in another train of length L2 meters, travelling at a speed of S2 m/sec in the opposite direction is L1/(S1 + S2) sec

9) Time taken by a train of length L1 meters, travelling at a speed of S1 m/sec to cross another train of length L2 meters, travelling at a speed of S2 m/sec in the same direction is (L1 + L2)/(S1 - S2) sec where S1 > S2

10) Time taken by a train of length L1 meters, travelling at a speed of S1 m/sec to cross another train of length L2 meters, travelling at a speed of S2 m/sec in the opposite direction is (L1 + L2)/(S1 + S2) sec

Q1) A train takes 9 sec to cross a pole. If the speed of the train is 48 km/hr, then length of the train is
1) 150 m
2) 120 m
3) 90 m
4) 80 m

Solution:

Train takes 9 sec to cross a pole
Length of the train = 48×(5/18)×9 = 120 m
Hence, option 2.

Q2) 150 m long train running with the speed of 90 km/hr crosses a bridge in 26 sec. What is the length of the bridge?
1) 500 m
2) 600 m
3) 650 m
4) 550 m

Solution:

Let the length of the bridge be y m.
Distance travelled = Length of the train + Length of the bridge = 150 + y
150 + y = 90×(5/18)×26
y = 500 m
Hence, option 1.

Q3) A train passes two persons who are walking in the direction opposite to the direction of train at the rate of 10 m/sec and 20 m/sec respectively in 12 sec and 10 sec respectively. Find the length of the train.
1) 500 m
2) 900 m
3) 400 m
4) 600 m

Solution:

Let the speed of the train be s m/sec.
(s + 10)×12 = (s + 20)×10
On solving, s = 40 m/sec
Length of the train = (s + 10)×12 = 600 m
Hence, option 4.

Q4) A train 110 m long travels at 60 km/hr. How long does it take to cross another train 170 m long, running at 54 km/hr in the same direction?
1) 2 min 40 sec
2) 2 min 48 sec
3) 3 min 48 sec
4) 3 min 40 sec

Solution:

Distance travelled = Length of train 1 + Length of train 2 = 110 + 170 = 280 m
As trains are travelling in the same direction so relative speed is difference of the speeds of the trains = 60 – 54 = 6 km/hr = 6×(5/18) = 5/3 m/sec
280 = 5/3 t
T = 168 sec = 2 min 48 sec
Hence, option 2.

Q5) Two trains are moving in opposite direction at 30 km/hr and 24 km/hr. The faster train crosses a man in the slower train in 6 sec. Find the length of the faster train.
1) 80 m
2) 100 m
3) 110 m
4) 90 m

Solution:

As trains are travelling in the opposite direction so relative speed is sum of the speeds of the trains = 30 + 24 = 54 km/hr = 15 m/sec
As faster train crosses a man in the slower train so,
Distance travelled = Length of the faster train = 15(6) = 90 m
Hence, option 4.

Q6) 250 m long train crosses a platform of length 350 m in 50 sec. Find the time of the train to cross a bridge of 230 m.
1) 45 sec
2) 50 sec
3) 40 sec
4) 54 sec

Solution:

Let s m/sec be the speed of the train
Distance travelled = Length of train + Length of platform = 250 + 350 = 600 m
600 = 50s i.e. s = 12 m/sec
Let t sec be the time taken by the train to cross the bridge of 230 m
Distance travelled = Length of train + Length of bridge = 250 + 230 = 480 m
480 = 12t i.e. t = 40 sec
Hence, option 3.

Q7) 2 trains of same length but with different speeds pass a static pole in 5 sec and 6 sec respectively. In what time will they cross each other when they are moving in the same direction?
1) 45 sec
2) 50 sec
3) 1 min
4) 90 sec

Solution:

As ratio of time is 5:6 so, ratio of speeds is 6:5
Speed of train 1 and train 2 be 6y, 5y respectively.
Length of the train = 6y(5) = 30y
Time taken to cross each other = 60y/y = 60 sec = 1 min
Hence, option 3.

Q8) A goods train and a passenger train are running on a parallel track in the same direction. The driver of the goods train observes that the passenger train coming from behind overtakes and crosses his train completely in 30 sec whereas a passenger on the passenger train marks that he crosses the goods train in 20 sec. If the speeds of the trains be in the ratio of 1:2, find the ratio of their lengths
1) 3:2
2) 3:1
3) 2:1
4) 4:1

Solution:

Length of goods train and a passenger train (in meters) is g, p respectively and their respective speeds be a, b.
As the two trains are running in the same direction so, distance travelled = Length of goods train + Length of passenger train = g + p
As the passenger train overtakes the goods train in 30 seconds so,
g + p = (b – a) 30      equation 1
Passenger on the passenger train crosses the goods train in 20 sec so, distance travelled = Length of the goods train
g = (b – a) 20      equation 2
On solving equation 1, 2
g/p = 2/1
Hence, option 3.