# Two trains running in opposite directions at 36 kmph and 45 kmph cross each other in 20 seconds. If one train is 200m long, what is the length of the other train?

(a). 144 m

(b). 200 m

(c). 240 m

(d). 250 m

Answer

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Hint: Find the relative velocity between the two trains. The total distance covered is the sum of lengths of both the trains. Then use formula for velocity to determine the sum of lengths of the trains. Since, the length of one train is known, the other can be found easily.

Complete step-by-step answer:

The two trains are running in opposite directions, the relative speed of one with respect to another is then the sum of their speeds.

Let v be the relative speed between the two trains, then v is given by:

\[v = 36 + 45\]

\[v = 81kmph\]

We know that \[1kmph = \dfrac{5}{{18}}m/s\] , converting v to m/s, we have:

\[v = 81 \times \dfrac{5}{{18}}m/s\]

\[v = \dfrac{{45}}{2}m/s{\text{ }}.........{\text{(1)}}\]

The total distance for the trains to completely cross each other is the sum of the lengths of the two trains. Let the total distance be d and the length of one train be \[{L_1}\] and the length of the other train be \[{L_2}\] . Then we have:

\[d = {L_1} + {L_2}{\text{ }}.........{\text{(2)}}\]

We know the formula for the total distance covered, when total time and the speed is given. It is given by:

\[d = v \times t\]

The time taken to cross each other is 20 seconds. Substituting equation (1), we have,

\[d = \dfrac{{45}}{2} \times 20\]

\[d = 45 \times 10\]

\[d = 450m{\text{ }}.........{\text{(3)}}\]

Using equation (3) in equation (2), we have:

\[{L_1} + {L_2} = 450\]

It is given that the length of one train is 200m, hence \[{L_1} = 200m\] . Substituting this in the above equation, we get

\[200 + {L_2} = 450\]

\[{L_2} = 450 - 200\]

\[{L_2} = 250m\]

Hence, the length of the other train is 250 m.

Therefore, the correct answer is option (d).

Note: The possibility of the mistake is that you can take the relative speed to be the difference between the two trains, which is wrong. When the trains are moving in opposite directions, the relative speed is the sum of both the speeds. You can also make an error by taking the total distance as the difference between the lengths of the trains but for the trains to cross them completely, we need to take the sum of the lengths of the trains.

Complete step-by-step answer:

The two trains are running in opposite directions, the relative speed of one with respect to another is then the sum of their speeds.

Let v be the relative speed between the two trains, then v is given by:

\[v = 36 + 45\]

\[v = 81kmph\]

We know that \[1kmph = \dfrac{5}{{18}}m/s\] , converting v to m/s, we have:

\[v = 81 \times \dfrac{5}{{18}}m/s\]

\[v = \dfrac{{45}}{2}m/s{\text{ }}.........{\text{(1)}}\]

The total distance for the trains to completely cross each other is the sum of the lengths of the two trains. Let the total distance be d and the length of one train be \[{L_1}\] and the length of the other train be \[{L_2}\] . Then we have:

\[d = {L_1} + {L_2}{\text{ }}.........{\text{(2)}}\]

We know the formula for the total distance covered, when total time and the speed is given. It is given by:

\[d = v \times t\]

The time taken to cross each other is 20 seconds. Substituting equation (1), we have,

\[d = \dfrac{{45}}{2} \times 20\]

\[d = 45 \times 10\]

\[d = 450m{\text{ }}.........{\text{(3)}}\]

Using equation (3) in equation (2), we have:

\[{L_1} + {L_2} = 450\]

It is given that the length of one train is 200m, hence \[{L_1} = 200m\] . Substituting this in the above equation, we get

\[200 + {L_2} = 450\]

\[{L_2} = 450 - 200\]

\[{L_2} = 250m\]

Hence, the length of the other train is 250 m.

Therefore, the correct answer is option (d).

Note: The possibility of the mistake is that you can take the relative speed to be the difference between the two trains, which is wrong. When the trains are moving in opposite directions, the relative speed is the sum of both the speeds. You can also make an error by taking the total distance as the difference between the lengths of the trains but for the trains to cross them completely, we need to take the sum of the lengths of the trains.

Last updated date: 28th Sep 2023

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