Trains

In this article, we will cover the various kinds of Time, Speed, Distance (TSD) questions that are based on the theme of Trains.

Type 1: Time taken by the train to cross a stationary object

Stationary object of negligible length

Time taken by a train of length ‘L’ meters to pass a stationary object of negligible length (e.g. a telegraph post or standing man) is equal to the time taken by the train to cover ‘L’ meters.

Time = $\frac{Length \hspace{1ex} of \hspace{1ex} train (L)}{Speed \hspace{1ex} of \hspace{1ex} train (S)}$

Q. In what time will a 720 metre train travelling at a speed of 36 km per hour cross a pole?

Explanation:

Stationary object of some length

Time taken by a train of length ‘L’ meters to pass a stationary object of length ‘b’ meters is –

T = $\frac{Length \hspace{1ex} of \hspace{1ex} train (L) + Length \hspace{1ex} of \hspace{1ex} object (b)}{Speed \hspace{1ex} of \hspace{1ex} train (S)}$

Q. In what time will a 720 metre train travelling at a speed of 36 km per hour cross a 180 metre long platform?

Explanation:

Type 2: Time taken by the train to cross a moving object

Moving object of negligible length

Now let us consider the cases wherein a train crosses a moving object of negligible length.

Here, two cases may arise:

Case 1

When object and train move in the same direction, T = $\frac{L}{|S_1 − S_2|}$

Where, L = length of train; $S_1$ = Speed of train; $S_2$ = Speed of object

Case 2

When object and train move in the same direction, T = $\frac{L}{S_1 + S_2}$

Where, L = length of train; $S_1$ = Speed of train; $S_2$ = Speed of object

Q. In what time will a 450 metre train travelling at a speed of 72 km per hour cross a man moving at a speed of 18 km/hr:

A. in the same direction as that of the train?
B. in opposite direction to that of the train?

Explanation:

Moving object of some length

We often encounter such a scenario when two trains are involved.

Here, two cases may arise:

Case 1

If two trains are of length $L_1$ m and $L_2$ m are moving in opposite directions at $S_1$ m/s and $S_2$ m/s, then:

Relative speed = $(S_1 + S_2)$ km/h

The time taken by the trains to cross each other, $t_1$ = $\frac{L_1 + L_2}{S_1 + S_2}$ sec

Case 2

If two trains of length $L_1$ m and $L_2$ m are moving in the same direction at $S_1$ m/s and $S_2$ m/s, then:

Relative speed = |$S_1 – S_2$| km/h

The time taken by the faster train to cross the slower train, $t_2$ = $\frac{L_1 + L_2}{|S_1 - S_2|}$ sec

Q. Two trains of lengths 300 m and 400 m have speeds of 54 km/hr and 72 km/hr respectively. In how much time will the trains completely pass each other, given that they are moving in:

A. the opposite direction?
B. the same direction with the faster train just on the tail of the slower train?

Explanation:

Another Concept

Now, if the two times are given (i.e. time taken to cross while travelling in the same and opposite direction), then we can find out the speeds of the trains too.

If $S_1$ is the speed of the faster train and $S_2$ is the speed of the slower train, then

Speed of faster train, $S_1$ = $(\frac{L_1 + L_2}{2})(\frac{1}{t_1} + \frac{1}{t_2})$

Speed of slower train, $S_2$ = $(\frac{L_1 + L_2}{2})(\frac{1}{t_1} - \frac{1}{t_2})$

$t_1$ – time taken by trains to cross each other if they travel in opposite directions and
$t_2$ - time taken by the faster train to cross slower train if they travel in the same direction.

Q. Two trains of lengths 300 m and 400 m take 20 seconds to cross each other when moving in opposite directions. The same trains take 140 seconds to cross each other when they are moving in the same direction. Find the speeds of the two trains.

Explanations :
1 2

Type 3: Stoppages

Q. Excluding stoppages, the speed of a train is 60 kmph and including stoppages, it is 54 kmph. For how many minutes does the train stop per hour?

(a) 10 minutes   (b) 8 minutes    (c) 15 minutes   (d) 6 minutes

Explanation: