Question

A train $120m$ is going towards the north direction at a speed of $8m{{s}^{-1}}$. A parrot flies at a speed of $4m{{s}^{-1}}$ towards the south direction parallel to the railway track. The time taken by the parrot to cross the train is
$\begin{align}
  & \text{A}\text{. }30s \\
 & \text{B}\text{. }15s \\
 & \text{C}\text{. }10s \\
 & \text{D}\text{. }5s \\
\end{align}$

Answer 1

- Hint: There are some conditions where one or more objects are moving in a frame which is non-stationary with respect to an observer. In these cases, we apply the concept of Relative velocity. We can assume one part of the system at rest and can calculate the velocity of the other part with respect to the part at rest.

Complete step-by-step solution
The relative velocity is defined as the velocity of an object or observer in the rest frame of another object or observer.
$\overrightarrow{{{V}_{AC}}}=\overrightarrow{{{V}_{AB}}}+\overrightarrow{{{V}_{BC}}}$
Where,
${{V}_{AB}}$ is the velocity of body A with respect to body B, ${{V}_{BC}}$ is the velocity of body B with respect to body C and ${{V}_{AC}}$ is the velocity of body A with respect to body C.
Let’s say the velocity of train is ${{v}_{t}}$ , the velocity of parrot is ${{v}_{p}}$, and the velocity of ground is ${{v}_{g}}$

According to the formula for relative velocity,
The velocity of parrot with respect to train is given as,
${{v}_{\dfrac{p}{t}}}={{v}_{\dfrac{p}{g}}}+{{v}_{\dfrac{g}{t}}}$
As the ground is at rest,
${{v}_{g}}=0$
If we consider the train at rest, the velocity of parrot with respect to the frame of train is,
${{v}_{\dfrac{p}{t}}}={{v}_{p}}-\left( -{{v}_{t}} \right)$
The negative sign signifies that the train and parrot are moving in opposite directions.
${{v}_{\dfrac{p}{t}}}={{v}_{p}}+{{v}_{t}}$
Given that,
$\begin{align}
  & {{v}_{p}}=4m{{s}^{-1}} \\
 & {{v}_{t}}=8m{{s}^{-1}} \\
\end{align}$
Therefore,
${{v}_{\dfrac{p}{t}}}=4+8=12m{{s}^{-1}}$
Now,
If the train is assumed at rest, the distance travelled by parrot will be equal to the length of the train, while the velocity will be the velocity of parrot with respect to train.
Time taken by parrot is the ratio of distance travelled by parrot to the speed of parrot
$T=\dfrac{D}{V}$
Where,
$D$ is the distance travelled by parrot, that is, $120m$
$V$ is the relative velocity (Speed) of parrot with respect to train, that is, $12m{{s}^{-1}}$
Putting values,
$\begin{align}
  & T=\dfrac{D}{V}=\dfrac{120}{12} \\
 & T=10s \\
\end{align}$
The time taken by the parrot to cross the train is $10s$
Hence, the correct option is C.

Note: One of the very basic life examples of our encounter with relative velocity is that we are sitting in a train and we see another train moving off and feel we are moving even though we are stopped at the platform. This type of illusion occurs because there is no way to distinguish between the uniform motion and being stationary.