Problems based on trains are same as the problems related to Speed, Time and distance.

Formula to convert km/h to m/s :-

We know that, 1 km = 1000 meters and 1 hour = 60 min = 60 × 60 = 3600 seconds

Then, 1 km/h =	1000 m	=	10 m
	3600 sec		36 sec

=	5	m/s
	18

∴ 1 km/h =	5	m/s
	18

From above it is clear that we convert km/h into m/s it should be multiply by 5/18.

Ex- Convert 540 km/h into m/s.

Solution:- 540 km/h = 540 ×	5	m/s = 30 × 5 = 150 m/s
	18

Formula to convert m/s to km/h :-

We know that, 1 meter =	1	km
	1000

and 1 second =	1	h
	3600

Then, 1 m/s =	1 / 1000	=	3600
	1 / 3600		1000

=	36	=	18
	10		5

From above it is clear that we convert km/h into m/s it should be multiply by 18/5.

Ex- Convert 50 m/s into km/h.

Solution:- 50 m/s = 50 ×	18	= 10 × 18 = 180 km/h
	5

Some useful Shortcut Methods ;-

1. When a train to cross a pole, a person or a signal post or any other object, then distance covered by train is equal to the length of the train.

Ex- A train covers 120 m in passing a standing man. Find the length of the train.
Solution:- we know that, when a train passing a standing man, then the distance covered by train is equal to the length of the train.
∴ Length of the train = 120 m.

Ex- A 120 m long train passes a signal post in 9 sec. Find the speed of the train in km/h.
Solution:- Given, length of train = 120 m
time taken = 9 sec

∴ Speed of train =	Distance covered	=	120
	time taken		9

=	40	m / s
	3

=	40	×	18	= 8 × 6 = 48 km/h
	3		5

Ex- A train takes 12 seconds to cross a pole. If the speed of the train is 72 km/h, then find the length of the train.

Solution:- Given, speed of train = 72 km/h = 72 × =	5	= 20 m / s
	18

time = 12 seconds
∴ Length of train = speed of train × time
= 20 × 12 = 240 meters

2. When a train cross a stationary object having some length like bridge, platform etc. then the distance covered by train is equal to the sum of the length of train and length of that particular stationary object.

Ex- 200 m long train running with the speed of 108 km/h to cross a bridge in 30 seconds. Find the length of the bridge.
Solution:- Given , Length of train = 200 m
time taken = 30 seconds

speed of train = 108 km/h = 108 ×	5	=  30 m / s
	18

Let the length of the bridge be x m.
Distance covered = speed of train × time
or, 200 + x = 30 × 30
or, 120 + x = 900
or, x = 900 - 200 = 700
∴ Length of bridge = 700 m.

Ex- A train travelling at a speed of 40 m/s crosses a 500 m long platform in 20 seconds. Find the length of the train .
Solution:- given, speed = 40 m/s Length of platform = 500 m
time = 20 seconds
Let the length of train be x m.
Distance covered = speed of train × time
or, x + 500 = 40 × 20
or, x + 500 = 800
or, x = 800 - 500 = 300
∴ Length of train = 300 m.

3. When two trains are moving in same direction, then their relative speed is equal to the difference of the speeds of both trains.
Let the speed of first train be s₁ km/h and speed of second train be s₂ km/h, then

Ex- Two trains are moving in the same direction with speeds 40 km/h and 50 km/h, respectively. Find the relative speed.
Solution:- Relative speed = 50 - 40 = 10 km/h

Ex- Two trains are moving in same direction with speeds of 30 km/h and 37 km/h, respectively. What is the speed of trains in respect of each other?
Solution:- Given, speed of first train = 30 km/h and
speed of second train = 37 km/h
∴ Relative speed = 37 - 30 = 7 km/h.

4. When two trains are moving in opposite directions, then their relative speed is equal to the sum of the speeds of both trains.
Let the speed of first train be s₁ km/h and speed of second train be s₂ km/h, then

Ex- Two trains are moving in opposite directions with speeds of 8 m/s and 14 m/s, respectively. Find their relative speed.
Solution:- Given, speed of first train = 8 m/s and
speed of second train = 14 m/s
∴ Relative speed = 8 + 14 = 22 m/s.

5. When two trains of length l₁ km and l₂ km are moving in the same direction with speeds of s₁ km/h and s₂ km/h respectively, then time taken to cross each other.
Where, s₁ > s₂

Ex- Two trains of lengths 100 m and 120 m are moving in same direction at 12 m/s and 8 m/s, respectively. Find the time taken by trains to cross each other.
Solution:-Given, length of first train = 100 m
length of second train = 120 m
speed of first train = 12 m/s
speed of second train = 8 m/s

Time taken =	l1 + l2
	s1 - s2

=	100 + 120	=	220	= 55 seconds
	12 - 8		4

Ex- Two trains of equal length are running on parallel lines in the same direction at 46 km/h and 36 km/h. The faster train passes the slower train in 36 sec. Find the length of each train.
Solution:- given, Speed of first train = 46 km/h
Speed of second train = 36 km/h
Time taken = 36 sec.
Let the length of each train be x km.
Relative speed = 46 - 36 = 10 km/h

= 10 ×	5	=	25	= m/s
	18		9

Speed of train =	Distance
	time

⇒	25	=	( x + x )
	9		36

⇒	25	× 36 = 2x
	9

⇒ 100 = 2x

⇒ x =	100	= 50
	2

∴ Length of each train = 50 m

6. When two trains of length l₁ km and l₂ km are moving in the opposite directions with speeds of s₁ km/h and s₂ km/h respectively, then time taken to cross each other.

Ex- Two trains of lengths 200 m and 250 m are moving in same direction at 5 m/s and 10 m/s, respectively. Find the time taken by trains to cross each other.
Solution:- Given, length of first train = 200 m
length of second train = 250 m
speed of first train = 5 m/s
speed of second train = 10 m/s

∴ Time taken to cross each other =	l1 + l2
	s1 + s2

=	( 200 + 250 )	=	450	= 30 seconds
	( 5 + 10 )		15

Ex- A 300 m long train crosses a platform which is three times of its length, in 5 min. What is the speed of the train.
Solution:- Given, Length of train = 300 m
Length of platform = 3 × 300 = 900 m
Time taken = 5 min = 5 × 60 = 300 sec

Speed of train =	Distance covered
	time taken

=	( 300 + 900 )	=	1200	= 4 m/s
	300		300

7. When two trains start from two points A and B towards each other and after crossing each other, they take x and y time in reaching points B and A respectively, then the ratio of speed first train to second train

Ex- Two trains start at same time from points A and B towards each other and after crossing, they take 16 h and 9 h in reaching points B and A, respectively. Find the ratio of speeds of the 1st train to that of the 2nd train.
Solution:- given, x = 16 h and y = 9 h
∴ Speed of 1st train : Speed of 2nd train = √y : √x
= √9 : √16
= 3 : 4

/8. When two trains leave A for B at time t₁ and t₂ and travel with speeds x km/h and y km/h respectively, then the distance from P, where the two trains meet

Ex- Two trains leave kanpur for delhi at 8:00 pm and 8:30 pm respectively and travel at 60 km/h and 90 km/h How many kilometres from kanpur, they will the two trains meet?
Solution:- Here, x = 60 km/h, y = 90 km/h, t₁ = 8:00 pm and t₂ = 8 : 30 pm

∴ Distance =		x × y		× ( t2 - t1 )
		y - x

=		60 × 90		×	30
		90 - 60			60

=		60 × 90		×	1
		30			2

= 90 km

9. If two trains start moving towards each other from stations A and B with the speeds of x km/h and y km/h, respectively. When they meet each other, it is found that one train covers distance d more than that of another train, then,

Ex- From stations M and N, two trains start moving towards each other at speed 125 km/h and 75 km/h, respectively. When the two trains meet each other, it is found that one train covers 50 km more than another. Find the distance between M and N.
Solution:- Let trains meet after time t at a distance d from M.
Then, another train coming from M covers a distance of ( d + 50 )
For station M, distance covered by first train
d + 50 = 125t
d = 125t - 50 ….. (1)
For station N, distance covered by first train
d = 75t …….(2)
From Eq. (1) and (2), we get
125t - 50 = 75t
or, 125t - 75t = 50
or, 50t = 50
or, t = 50/50 = 1 h
∴ Distance between M and N = 125t + 75t = 200t = 200 × 1 = 200 km
By Formula,
Here, x = 125 km/h, y = 75 km/h and d = 50 km

Distance between M and N =		x + y		× d
		x - y

=	125 + 75	× 50 =	200	× 50
	125 - 75		50

= 200 km

10. When a train of length l m, passes a platform of p₁ m in t₁ sec, then time taken t₂ sec by the same train to pass a platform of length p₂ m is given as

Ex- A train of length 200 m, passes a platform of 250 m length in 15 sec. What time will this train take to pass the platform of 400 m length.
Solution:- Length of train = 200 m
Length of platform = 250 m
time taken = 15 sec.

∴ Speed of train =	Distance covered	=	( 200 + 250 )
	Time taken		15

=	450	= 30 m/sec
	15

Now, the time taken to cross the platform of length 400 m

∴ Time taken =	Distance covered	=	( 200 + 400 )
	speed		30

=	600	= 20 sec
	30

By Formula,
Here, l = 200 m, p₁ = 250 m, t₁ = 15 sec, p₂ = 400 m, t₂ = ?

∴ t2 =		l + p2		× t1
		l + p1

t2 =		200 + 400		× 15
		200 + 250

t2 =	600	× 15 = 20 sec.
	450

11. When a train travels without stoppage at an average speed of x km/h and with stoppage, it covers the same distance at an average speed of y km/h, then

Ex- Without stoppage, the speed of a train is 45 km/h and with stoppage, it is 36 km/h. For how many minutes, does the train stop per hour?
Solution:- Given, without stoppage speed = 45 km/h
With stoppage speed = 36 km/h
Decrease in speed due to stoppage = 45 - 36 = 9 km/h
Because of stoppage, train covers 9 km per hour.

∴ Time taken to cover 9 km =	9	=	1	h
	45		5

=	1	× 60 = 12 min.
	5

By Formula,
Here, x = 45 km/h and y = 36 km/h

Time of rest pert hour =		x - y
		x

=	( 45 - 36 )	=	9
	45		45

=	1	h =	1	× 60 = 12 min
	5		5

12. When two trains of equal lengths and different speeds take x and y time to cross a pole, then time taken by them to cross each other

When trains moves in opposite direction then used '+' sign and when moves in same direction used '-' sign.

Ex- Two trains of equal lengths take 8 sec and 10 sec respectively to cross a pole. If these trains are moving in the same direction. then how long will they take to cross each other?
Solution:- Given, Time taken by first train to cross a pole = 8 sec
Time taken by second train to cross a pole = 10 sec
Let the length of each train be x m.

Speed of first train =	Distance covered	=	x	m/sec
	Time taken		8

Speed of second train =	x	m/sec
	10

Now, time taken to cross each other =	Distance covered
	Speed

=	( x + x )
	{ ( x / 8 ) - ( x / 10 ) }

=	2x	= 80 sec.
	( x / 40 )

By Formula,
Here, x = 8 sec and y = 10 sec

∴ T =	2xy
	( y - x )

=	2 × 8 × 10
	( 10 - 8 )

T =	2 × 8 × 10	= 80 sec.
	2