Question

A train of length $200m$ traveling at a speed of $30m{s^{ - 1}}$ overtakes another train of length $300m$ traveling at a speed of $20m{s^{ - 1}}$ in the same direction. The time taken by the first train to pass the second train is:
A) $50s$
B) $10s$
C) $20s$
D) $40s$

Answer 1

Here it is given that two trains are travelling and one train overtakes another train, so it is clear that both the trains are moving in the same direction. Now in order to time that the first train takes to pass the second train the relative speed of both the trains and total distance travelled is needed. Calculate the relative speed of both the trains and also the total distance travelled and put both in the formula of time, distance and speed and you will get your answer.

Complete Step by Step Solution:
Here it is given that one train overtakes another train, this can only be possible when these two trains are moving in the same direction. It is anyway not possible that these two trains are moving in opposite directions.
Now the length of train one is given as, ${l_1} = 200m$
The speed with which train one is moving is, ${v_1} = 30m{s^{ - 1}}$
Now the length of train two is given as, ${l_2} = 300m{s^{ - 1}}$
The speed of train two is given by, ${v_2} = 20m{s^{ - 1}}$
Now as both the trains are moving in the same direction, the relative speed of the two trains would be subtraction of the given speeds of the two trains.
So, $v = {v_1} - {v_2}$
Putting the respective values we get,

$v = (30 - 20)m{s^{ - 1}}$

On solving we have,

$v = 10m{s^{ - 1}}$

Now as the trains cross each other the total distance travelled by train one to cross train two would be addition of lengths of both the trains.
So, total distance $d = {l_1} + {l_2}$
On putting the values of the respective lengths we have,

Total distance $d = (200 + 300)m$

On solving we have, Total distance $d = 500m$

Now the time taken would be given by total distance travelled divided by relative speed.

So, $t = \dfrac{d}{v}$

On putting the values we have,

$t = \dfrac{{500}}{{10}}$

On solving we have, $t = 50s$ .

So, the correct answer to this question is option A that is $50s$ .

Note: It is important to note the concept of relative speed. It is the speed of one object with respect to the other object in which one object is assumed to be in rest with respect to the frame of reference of another object. When both the objects are moving in the same direction the relative speed is the subtraction of speed of object one and object second while if they are moving in the opposite direction their speeds get added.