Question

An engine approaches a hill with a constant speed. When it is at a distance of $0.9km$, it blows a whistle whose echo is heard by the driver after 5 seconds. The speed of sound in air is $330m/s$, then the speed of the engine is:
(A) $32m/s$
(B) $27.5m/s$
(C) $60m/s$
(D) $30m/s$

Answer 1

Hint
We can find the distance travelled by the sound from the train by subtracting the distance travelled by the train in 5 seconds. Now dividing that distance by the time 5 seconds and equating it with the given speed of sound we can find the distance travelled by the train. From there we can find the speed of the train by dividing the distance by 5 seconds.
In this solution we will be using the following formula,
$\Rightarrow v = \dfrac{d}{t}$ where $v$ is the velocity, $d$ is the distance travelled and $t$ is the time taken.

Complete step by step answer
After the whistle is blown by the train the driver hears the echo after a time of 5 seconds. Let the distance travelled by the train during this 5 seconds be, $x$ meter.
Now the train is initially at a distance of $0.9km = 900m$ from the hill when it blows the whistle. So the sound wave travels a distance of $900m$ from the train to the hill. The echo of the sound wave returns from the hill to the train after 5 seconds. During those 5 seconds the train has travelled a distance of $x$ meter.
Therefore, the distance travelled by the sound wave from the hill to the train is $\left( {900 - x} \right)m$. Hence the total distance that the sound wave travels from the train to the hill and then back to the train is,
$\Rightarrow d = 900 + \left( {900 - x} \right)$
On opening the brackets we get
$\Rightarrow d = \left( {1800 - x} \right)m$
Now in the question it is given that the time taken by the sound wave to travel this distance is 5seconds.
Therefore the speed of the wave can be found by the formula,
$\Rightarrow v = \dfrac{d}{t}$
We are provided that the speed of the sound wave is $330m/s$. So substituting all these values we can write the equation as,
$\Rightarrow 330 = \dfrac{{1800 - x}}{5}$
On multiplying 5 on both the sides we get
$\Rightarrow 330 \times 5 = 1800 - x$
Now taking the $x$ to the LHS and the rest of the values to the RHS we get
$\Rightarrow x = \left( {1800 - 1650} \right)m$
Hence the distance travelled by the train is $x = 150m$
We know that the time taken to travel this distance is 5 seconds.
So again from the formula we have,
$\Rightarrow v = \dfrac{d}{t}$
Substituting the values we get
$\Rightarrow v = \dfrac{{150}}{5}m/s$
On calculation, we get the speed of the train as,
$\Rightarrow v = 30m/s$
So the correct answer will be option (D).

Note
The echo of a sound is the reflection of a sound that arrives to a listener with a little delay after the original sound. The object from which the sound gets reflected has to be at a minimum distance of $17.2m$ for the echo to be perceived by a person.