# Trains

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We know that when the direction of the movement of a boat and a river is the same, the relative speed is obtained by adding the speeds of both, the boat and the river. But if two trains are moving in the same direction, then what is the relative speed?

Let us see some cases:

Let us see some cases:

- When two trains of length L
_{1}and L_{2}and speed V_{1}m/s and V_{2}m/s respectively are crossing each other:- The direction of the movement of both the trains are the same:
_{1 }âˆ’ V_{2}|_{1 }+ L_{2} _{The direction of the movement of both the trains are opposite:}_{}_{Relative speed = | V1 + V2 |}_{Total distance covered = L1 + L2}

- The direction of the movement of both the trains are the same:
- When a train is crossing a stationary object:
- When the train is crossing a pole or a stationary human being:
_{1 }+ V_{2}, since V_{2 }= 0, then, the relative speed = V_{2}_{Total distance covered = L1 + L2, since L2 (width of the pole) is negligible with respect to L2 (Length of the train), so we do not consider it while calculating the quantities. Thus, distance = L1}_{However, it should be remembered that this is mathematically not correct and all the solutions are on the assumption that the width of the pole is zero, which is obviously not true.} _{When the train is crossing a platform or a standing train:}_{Relative speed = V1 + V2,}

- When the train is crossing a pole or a stationary human being:

Where V

Since V

Total distance covered = L

Where L

_{1}is the speed of the moving train and V_{2}is the speed of the standing train or the platform.Since V

_{2 }= 0, so the relative speed = V_{1}Total distance covered = L

_{1 }+ L_{2}Where L

_{1 }is the length of the moving train and L_{2 }is the length of the standing train or the platform.Example-1

A train takes 10 s to cross a pole and 20 s to cross a platform of length 200 m. What is the length of train?

Solution

The train takes 10 s to cross its own length and 20 s to cross its own length and length of the platform. So, it is infered that the train takes 10 s to cross the platform and 10 s to cross its own length.
Since the time taken to cross the platform = time taken to cross its own length
So, length of the platform = length of the train = 200 m

Example-2

Speed of a train is 36 km/h. It takes 25 s to cross a pole. What is the length of this train?

Solution

Speed of train = 10 m/s (36 Ã— 5/18)
Distance covered = 10 Ã— 25 = 250 m
So, length of train = 250 m

# Some Special Cases

**Case 1:**

**Two trains are moving in an opposite direction with a speed of V**

_{1 }and V

_{2}. Their lengths are L

_{1 }and L

_{2}. Now, see the whole situation from the point of view of a person sitting on the window seat of the 1st train.

_{1 }+ V

_{2}(This person can be assumed to be running with a speed of V

_{2.})

Relative distance = L

_{2}**Case 2:**

**A train is running with a speed V**

_{1}and a person X is running inside the train with a speed of V

_{2 }in the direction of the movement of train. Now if a person Y is watching this from the outside of the train, then the relative speed of Y with respect to X = V

_{1 }+ V

_{2.}

_{}Speed of person X with respect of another person Z who is sitting in the train = V

_{2}

_{}There is also a person P who is outside the train and is moving with a speed of V

_{3 }in the opposite direction of train.

X = V

_{1 }+ V_{2 }+ V_{3}_{}Had this person P been running in the same direction as that of the train, then the relative speed of P with respect to person X = | V_{1 }+ V_{2}âˆ’ V_{3 }|