Problems on Trains - Important Formulas
km/hr to m/s conversion:
m/s to km/hr conversion
Formulas for finding Speed, Time and Distance
Time = $\dfrac{distance}{speed}$
Distance = $Speed \times Time $
Relative speed
Formulas
Relative speed at opposite direction = (x + y) km/hr
Directions
Formulas
i) If two trains of length a metres and b metres are moving in opposite directions at u m/s
and v m/s, then:
The time taken by the trains to cross each other =$\left(\dfrac{a+ b}{u+v}\right)$
ii) 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 train to cross each other =$\left(\dfrac{a+ b}{u-v}\right)$
Ratio of the Speed
Formulas
The train based problems are based on two object, First is Train and second object is that which is crossed by the train.
Speed
The rate of change of distance with respect to time.
Formulas
Example
A train crossed 30 km in 5 hours. Find speed of the train.
here,
Distance covered = 30 km
Time taken = 5 hours
We know, speed = $\dfrac{distance}{time}$
= $\dfrac{30}{5}$ km/hr
Therefore, speed = 6 km/hr
Exercise
Speed of the train = 108km/h = 108 $\times \dfrac{5}{18}$ m/s = 30m/s
Speed =$\dfrac{distance}{time}$
Length of bridge + Train = $\dfrac {600}{30}$ = 20 seconds
Time
It is the time duration over which the movement has occurred.
Formulas
Example
The train of length 100m moving at a speed of 60km/hr will pass a man standing on the platform in,
Time to pass a stationary man
= $\dfrac{100m}{60km/hr}$
=$\dfrac{100m}{60000m/hr}$
=$\dfrac{3600}{600}secs$
=6secs .
Exercise
Formula::Time = $\dfrac{distance} {speed}$.
Speed of the train = 216km/h = 216 $\times \dfrac{5}{18}$ m/s = 60m/s
Length of bridge + Train = 600 + 600 = 1200 m
= $\dfrac{1200}{60}$ = 20 seconds
Distance
Total area covered with in respect of time.
Formulas
Example
A train is running at a speed of 20 m/sec.. If it crosses a pole in 30 seconds, find distance of the train to cross the pole.
here,
Speed = 20 m/sec
Time = 30 seconds
Distance=Speed*Time
= 20 $\times$ 30
= 600 meters
Exercise
Distance=Speed*Time
= 10 $\times$ 15
= 150 meters
Relative speed
The concept of relative Speed is used when two or more trains moving with some Speeds are considered
Formulas
Relative speed at opposite direction = (x + y) km/hr
Example
Find the relative speed of Train A travelling at 120km/h with respect to Train B is 70km/h in same direction
Train A is travelling at 120 km/h
Train B is 70km/h
Relative speed = (x - y) km/h.
= 120 - 70
= 50km/h
Exercise
Train A is travelling at 120 km/h
Train B is 70km/h
Relative speed = (x + y) km/h.
= 120 + 70
= 190km/h
Directions
Formulas
ii) 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 train to cross each other =$\left(\dfrac{a+ b}{u-v}\right)$
Example
Two trains of equal lengths take 10 seconds and 15 seconds respectively to cross a telegraph post.
If the length of each train be 120 metres in what time (in seconds) will they cross each other travelling in opposite direction?
Speed of the first train = $\left(\dfrac{120}{10} \right)$ m/sec = 12 m/sec.
Speed of the second train =$\left(\dfrac{120}{15} \right)$ m/sec = 8 m/sec.
Relative speed = (12 + 8) = 20 m/sec.
∴ Required time =$[\dfrac{(120 + 120)}{20}]$ sec = 12 sec.
Exercise
| Relative speed = (40 - 20) km/hr =$ \left(20 \times\dfrac{5}{18} \right) $m/sec =$ \left(\dfrac{50}{9} \right) $m/sec. |
| $\therefore$ Length of faster train =$ \left(\dfrac{50}{9} \times 5\right) $m =$ \dfrac{250}{9} $m = 27$ \dfrac{7}{9} $m. |
Ratio of the Speed
Formula:
Example
Two trains one from Howrah to Patna and the other from Patna to Howrah start simultaneously. After they meet the trains reach their destinations after 9 hours and 16 hours respectively. The ratio of their speeds is:
Let us name the trains as A and B. Then,
A's speed : B's speed = $\sqrt{b} : \sqrt{a} = \sqrt{16} : \sqrt{9}$
= 4 : 3