Question

A train approaching a hill at a speed of $40\dfrac{{km}}{{hr}}$ sounds a whistle of frequency 600 Hz when it is at a distance of 1 km from a hill. A wind with a speed of $40\dfrac{{km}}{{hr}}$ is blowing in the direction of motion of the train. The distance from the hill at which the echo from the hill is heard by the driver and its frequency (velocity of sound in air $= 1200\dfrac{{km}}{{hr}}$).

Answer 1

We will calculate distance travelled by sound wave to reach the hill and return back to the hill. We will calculate time taken in covering that distance. Using,
$Speed = \dfrac{{Dis\tan ce}}{{Time}}$
we will calculate the distance of echo. Then the frequency of the sound wave is calculated using formula.
$f'' = f'\left( {\dfrac{{v' - {v_0}}}{{v'}}} \right)$

Complete step by step answer:
Let us assume the speed of the sound in air is ‘c’ and speed of air is ‘${v_a}$’.
Sound is moving in the direction of speed of air;
$\Rightarrow$ $v = c + {v_a}$
Let us assume ‘x’ be the distance of the train from the hill when it blows a whistle.
As we know,
$\Rightarrow$ $Speed = \dfrac{{Dis\tan ce}}{{Time}}$
Let ‘v’ be the speed of the sound coming from distance ‘x’ in ‘t’ time.
Then, $v = \dfrac{x}{t}$
$\Rightarrow$ $t = \dfrac{x}{{c + {v_a}}}$ … (1)
When sound is reflected back, sound is moving in opposite direction of speed of air;
$\Rightarrow$ $v = c - {v_a}$
Let ‘$x'$’ be the distance travelled by reflected sound wave, and ‘$t'$’ be the time taken to travel reflected wave
$\Rightarrow$ $t' = \dfrac{{x'}}{{v'}}$
$\Rightarrow$ $t' = \dfrac{{x'}}{{c - {v_a}}}$ … (2)
Total time taken $= t + t'$
Total distance travelled by train $= x - x'$
It is moving with constant speed ‘${v_s}$’
$\Rightarrow$ $t + t' = \dfrac{{x - x'}}{{{v_s}}}$
$\Rightarrow$ $x - x' = (t + t')\,{v_s}$
Using equation (1) and (2),
$\Rightarrow$ $x - x' = \dfrac{x}{{c + {v_a}}}{v_s} + \dfrac{{x'}}{{c - {v_a}}}\,{v_s}$
$\Rightarrow x\dfrac{{c + {v_a} - {v_s}}}{{c + {v_a}}} = \,x'\dfrac{{c - {v_a} + {v_s}}}{{c - {v_a}}}$
$\Rightarrow x' = x\dfrac{{(c - {v_a}) \times (c + {v_a} - {v_s})}}{{(c + {v_a}) \times (c - {v_a} + {v_s})}}$
$\Rightarrow x' = \dfrac{{(1200 - 40) \times (1200 + 40 - 40)}}{{(1200 + 40) \times (1200 - 40 + 40)}}$
$\Rightarrow x' = 0.935\,km$
$\Rightarrow x' = \,935m$
Echo will be heard when the train is at a distance of 935m away from the hill.
Source I.e., hill is stationary and an observer is moving towards the hill travelling at a speed of $40\dfrac{{km}}{{hr}}$.
Reflected sound waves travel in the opposite direction to the wind.
Velocity of echo
$\Rightarrow$ $v' = 1200 - 40 = 1160km$
Frequency of sound coming from hill
Distance from the sound wave is 935 m and frequency is 641 Hz.

Additional information:
Echo is an effect observed when the observer can distinctly hear a reflection of the sound that was created by the observer themselves.
For humans the reflecting surface should be 17 m away so as to distinctly hear the echo.

Note:
In this case no conversion of units is required, unnecessary conversions might lead to wrong solutions.