Question

A train travelling at $250\,Km/hr$ overtakes a cyclist who is moving at $10\,Km/hr$ in $45$ seconds. Find the length of the train in metres?

Answer 1

We first find the relative speed of the train. In crossing the cyclist, the train actually crosses its own length. We convert the units from $km/hr$ to $m/\sec$ by multiplying with $\dfrac{5}{18}$. Then we multiply with the time 45 to find the final solution.

Complete step by step answer:
The train and the cyclist are running in the same direction at the speed of $250km/hr$ and $10km/hr$ respectively. It is also given that the train passes the cyclist in 45 seconds.The concept is of passing a moving object with respect to an object of negligible length. In that case the moving object actually passes the length of itself.

Now as the train and the cyclist are running in the same direction, the speed of the train at which it crosses the cyclist will be the relative speed. Relative speed will also be the difference of the speeds as they are moving in the same direction which is $250-10=240\,km/hr$.

It crosses the cyclist in 45 seconds. The relative speed is 240 kilometres in 1 hour.
We can transform the units from $km/hr$ to $m/\sec$ by multiplying with $\dfrac{5}{18}$.
Therefore, $240\,km/hr$ is equal to $240\times \dfrac{5}{18}=\dfrac{200}{3}\,m/\sec$.
In 45 seconds, it will cover $\dfrac{200}{3}\times 45=3000$ metres which is the length of the train.

Therefore, the length of the train is 3000 metres or 3 Km.

Note: In case the trains go in the opposite direction, the relative speed would have been the sum of the speeds instead of the subtraction. The speed of separation is the cumulative speed of those moving objects.