Question

The rails on a railroad are 30 m long. As the train passes over the point where the rails are joined, there is an audible click. The speed of the train in km per hour is approximately the number of clicks heard in
A.108 seconds\[\]
B. 2 minutes\[\]
C. $1\dfrac{1}{2}$ minutes\[\]
D. 5 minutes\[\]
E. None of these \[\]

Answer 1

Hint: We assume our position of observation at the tail end of the train. When the train will pass the rail it will also pass the tail end. We assume the speed of the train is $x$ km/hr. We convert it into m/s and then multiply with the number of clicks heard per meter. We find the number of clicks we hear in a second whose reciprocal will be the result.

Complete step-by-step solution:
We are given the question that the rails on a railroad are 30 m long. We hear an audible click when the train passes over the point where the rails are joined. \[\]
Let us assume that we are at the tail end of the train and we hear the click. The train has to pass its own length to reach the end of the train where we are positioned to hear the click. .We are also given the data that the number of clicks we hear to after the end of the train crosses the rail is approximately equal to the speed of the train.\[\]
So let us assume that the speed of the train is $x$ kilometre per hour but we are given the length of the rail as a meter. We convert 1 kilometer to meter and get 1km=1000m. We convert 1 hour to seconds and get 1hour=3600 minutes. So the speed of the train in meter per second m/s is
\[x\text{ km/hr=}x\times \dfrac{1000m}{3600\text{s}}=x\times \dfrac{5}{18}\text{m/s}\]
 We hear a click when the train passes a rail or in other words, covers a distance of 30m. No of clicks per metre is $\dfrac{1}{30}$. So the number of clicks per second is
\[x\times\dfrac{5}{18}\text{m/s}\times \dfrac{1\text{click}}{30m}=\dfrac{5x}{5400}\text{click/s}\]
The question asks for how many seconds $\dfrac{5x}{5400}$ audible clicks are heard. Let us assume we hear the clicks for $y$ seconds. So the number of clicks we heard in $y$ seconds is $y\times \dfrac{5x}{5400}$ which according to the question is equal to the speed of the train. We have taken the speed of the train here is $x.$ So we have,
\[\begin{align}
 & y\times \dfrac{5x}{5400}=x \\
 & \Rightarrow y=\dfrac{5400}{5}=108 \\
\end{align}\].
So the speed of the train is the number of audible clicks heard in 108 seconds. So the correct option is A.

Note: We note that if the length of the train is $l$ and the length of the rail is $b$ and if the train crosses the rail at a speed $v$ then the time taken to cover one end of the rail to other is $t=\dfrac{l+b}{v}$. The time taken by the train to signal post is $t=\dfrac{l}{v}$.