Question

A soldier walks forwards a high wall taking $120$ steps per minute. When he is at a distance of $90\,m$ from the wall he observes that echo of step coincides with the next step. The speed of sound must be
$1)$ $340\,m/s$
$2)$ $330\,m/s$
$3)$ $300\,m/s$
$4)$ $360\,m/s$

Answer 1

Hint:We know the value of distance travelled per unit of time in this above statement and then find the speed of the sound using the simple principle of measuring speed and distance. The speed of sound is the distance travelled by a sound wave per unit of time. The time interval is determined.

Useful formula:
${\text{Distance = Speed}}\,{\text{of}}\,{\text{the}}\,{\text{sound}} \times {\text{Time}}\,{\text{taken}}$
\[{\text{Speed of}}\,{\text{the sound = }}\dfrac{{{\text{distance travelled}}}}{{{\text{time}}\,{\text{taken}}}}\]
${\text{Time}}\,{\text{taken = }}\dfrac{{{\text{Distance}}\,{\text{travelled}}}}{{{\text{Speed}}\,{\text{of}}\,{\text{the}}\,{\text{sound}}}}$
Units of this formula given below,
Time: \[{\text{seconds}}\left( {\text{s}} \right){\text{, minutes }}\left( {{\text{min}}} \right){\text{, hours }}\left( {{\text{hr}}} \right)\]
Distance: \[{\text{meters }}\left( {\text{m}} \right){\text{, kilometers }}\left( {{\text{km}}} \right){\text{, miles, feet}}\]
Speed: \[{\text{m/s, km/hr}}\]
One minute is equal to $60\,\sec $

Complete step by step solution:
Given by,
Number of steps $ = 120$
Time taken $ = 60\sec $
Distance traveled by the sound wave $ = 90\,m$
Now,
We applying the formula,
Number steps in one second $ = \dfrac{{120}}{{60}}$
Therefore, the number of steps in one second is $2$steps.
Total distance traveled by sound wave is $90 \times 2$$ = 180$
when echo previous step is coincide with the next step then time interval is \[1/2 = 0.5{\text{ }}s\]
(because $2$ steps are covered in $1$ second so one step will be covered in$1/2\,s$)
We know that,
Speed formula,
\[{\text{Speed of}}\,{\text{the sound = }}\dfrac{{{\text{distance travelled}}}}{{{\text{time}}\,{\text{taken}}}}\]
Where,
Distance traveled in $180$$m$
Time taken is $0.5s$
Substituting the given value in above formula,
\[{\text{Speed of}}\,{\text{the sound }} = {\text{ }}\dfrac{{180}}{{0.5}}\]
The speed of the sound is $360\,m/s$
Hence,
The speed of sound must be $360\,m/s$

Therefore, the option $4)$ is the correct answer.

Note:In internal energy, the speed of sound depends only on its temperature and its composition. In ordinary air, the velocity has a poor dependency on frequency and pressure, slightly deviating from the observed value. which ensures that the sound travels quicker.