18. PROBLEMS ON TRAINS
IMPORTANT FACTS AND FORMULAE
1. a km/hr= (a* 5/18) m/s.
2. a m / s = (a*18/5) km/hr.
3 Time
taken by a train of length 1
metres to pass a pole or a standing man or a signal post is equal to the time taken by the
train to cover 1 metres.
4. Time taken by a train of length 1 metres to
pass a stationary object of length b metres
is the time taken by the train to cover (1 + 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 m / s and v m/s, then time taken
by the trains to cross each other = (a
+ b)/(u+v) sec.
8.If two trains
of length a metres and b metres
are moving in the same direction
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 two 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 speet) : (B’s speed) = (b1/2: a1/2).
SOLVED EXAMPLES
Ex.I. A train 100 m long is running at the speed of 30
km / hr. Find the time taken by it
to pass a man standing
near the railway line. (S.S.C. 2001)
Sol.
Speed of the train = (30 x 5/18_)
m / sec = (25/3) m/ sec.
Distance moved in passing the standing man = 100 m.
Required time taken = 100/(25/3) =
(100 *(3/25)) sec = 12 sec
Ex. 2. A
train is moving at a speed of 132 km/br. If the length of the train is
110 metres,
how long will it take
to cross a railway platform 165 metres long?
(Section
Officers', 2003)
Sol.
Speed of train = 132 *(5/18) m/sec = 110/3 m/sec.
Distance
covered in passing the platform = (110 + 165) m = 275 m.
Time taken =275 *(3/110) sec =15/2 sec = 7 ½ sec
Ex. 3. 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?
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
sec
x/8=(x+180)/20
ó 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 km
Ex. 4. A train 150 m long is running with a
speed of 68 kmph. In what
time will it pass a man
who is running at 8 kmph in the same direction in which the train is going?
Sol: Speed of the train relative to man = (68 - 8) kmph
= (60* 5/18) m/sec = (50/3)m/sec
Time taken by the train to cross the man I
= Time taken by It to cover 150 m at 50/3
m / sec = 150 *3/ 50 sec = 9sec
Ex. 5. A train 220 m long is running with a speed of 59
kmph.. In what will
it pass a man who is running at 7 kmph in
the direction opposite to that in which the train 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 220 m at (55/3) m / sec
= (220 *3/55) sec = 12 sec
Ex. 6. Two trains 137 metres and 163 metres
in length are running towards
each other on parallel
lines, one at the rate of 42 kmph and another at 48 kmpb. In what time will they be clear of each other from the
moment they meet?
Sol. Relative speed of the trains = (42 + 48) kmph = 90
kmph
=(90*5/18)
m / sec = 25 m /sec.
Time taken by the trains to'pass each other
= Time
taken to cover (137 + 163) m at 25 m /sec =(300/25) sec =
12 sec
Ex. 7. Two trains 100 metres and 120 metres long are running in the same direction with speeds of 72 km/hr,In howmuch time
will the first train cross the second?
Sol: Relative speed of the trains = (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.
Ex. 8. A train 100 metres long takes 6 seconds to cross a man
walking at 5 kmph in the direction opposite to that of the train. Find the
speed of the train.?
Sol:Let the speed of the train be x kmph.
Speed of the train relative to man = (x + 5) kmph = (x +
5) *5/18 m/sec.
Therefore 100/((x+5)*5/18)=6 <=> 30
(x + 5) = 1800 <=> x = 55
Speed of the train is 55 kmph.
Ex9. A train running at 54 kmph
takes 20 seconds to pass a platform. Next it takes.12 sec to pass a man walking at 6 kmph in the same
direction in which the train is going . Find the length of the train and the
length of the platform.
Sol:Let the length of train be x metres and
length of platform be y metres.
Speed of the train relative to man = (54 -
6) kmph = 48 kmph
= 48*(5/18)
m/sec = 40/3 m/sec.
In passing
a man, the train covers its own length with relative speed.
Length of
train = (Relative speed * Time) = ( 40/3)*12 m = 160 m.
Also,
speed of the train = 54 *(5/18)m / sec = 15 m / sec.
(x+y)/15
= 20 <=> x + y = 300 <=> Y = (300 - 160) m =
140 m.
Ex10. A man
sitting in a train which is traveling at 50 kmph observes that a goods train, traveling 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.
