Question

A train of 150 m length is going towards the north direction at a speed of 10 m/s. a parrot flies at a speed of 5 m/s towards south direction parallel to the railway track. The time taken by the parrot to cross the train is equal to:

Answer 1

Hint: We need to find the relative speed of the bird with respect to the train. The magnitude of a velocity of an object can be given simply the magnitude of the displacement divided by the time taken to cover such distance.
Formula used: In this solution we will be using the following formulae;
\[{v_{12}} = {v_{1g}} - {v_{2g}}\] where \[{v_{12}}\] is the speed of an object 1 relative to object 2, \[{v_{1g}}\]is the velocity of object 1 relative to the stationary ground, and \[{v_{2g}}\] is the velocity of the object 2 relative to the stationary ground (all velocities are in vector form).
\[\left| v \right| = \dfrac{{\left| d \right|}}{t}\] where \[\left| v \right|\] is the magnitude of the velocity, \[\left| d \right|\] is the magnitude of the displacement covered, and \[t\] is the time taken to cover the displacement.

Complete Step-by-Step Solution:
A train of a particular length is said to be moving at a particular velocity northward, and a bird is moving parallel to the train, we are to calculate how long it takes the bird to cross the train.
To do this, let’s assume that we are on the train. Velocity is relative, hence on the reference frame of the train, it as though itself is static while the bird is moving at a velocity with respect to it. This velocity is given by
\[{v_{bt}} = {v_{bg}} - {v_{tg}}\] where \[{v_{bt}}\] is the speed of the bird relative to the train , \[{v_{bg}}\]is the velocity of the bird as observed by person on the ground, and \[{v_{tg}}\] is the velocity of the train relative to the ground (all velocities are in vector form).
Hence, the velocity of bird relative to the train,
\[{v_{bt}} = - 5 - 10 = - 15m/s\] (assuming northward is positive)
Then, from \[\left| v \right| = \dfrac{{\left| d \right|}}{t}\] where \[\left| v \right|\] is the magnitude of the velocity, \[\left| d \right|\] is the magnitude of the displacement covered, and \[t\] is the time taken to cover the displacement, the time taken to cover the length of the train will be
\[t = \dfrac{{\left| d \right|}}{{\left| v \right|}} = \dfrac{{150}}{{15}}\]
\[ \Rightarrow t = 10s\]

Note: Alternatively, we could assume the bird was static, and the train crosses the bird. Then, the velocity of the train relative to bird is
\[{v_{tb}} = {v_{tg}} - {v_{bg}} = 10 - \left( { - 5} \right)\]
\[{v_{tb}} = 15m/s\]
This has the same magnitude as the velocity of the train relative to the bird as calculated in solution. Hence, the time taken would be equal.