Question

A construction worker’s helmet slips and falls down when he is 78.4 m above the ground. He hears the sound of the helmet hitting the ground 4.23 seconds after it has slipped. Find the speed of sound in air.

Answer 1

Step 1- Given Data

Height above the ground $h=78.4m.$

Time taken by the sound to be heard $t\text{'}=4.23s$

Acceleration due to gravity $g=9.8m/s^2$

Step 2- To find out the time taken by the helmet to reach the ground

According to second equation of motion,

$h=ut+\frac{1}{2}g{t}^2$

Where $u$ is the initial velocity and$t$is time taken.

$78.4=0+\frac{1}{2}\left(9.8\right)t^2$

$$\begin{gathered} \Rightarrow 78.4=\frac{1}{2}\left(9.8\right)t^2 \\ \Rightarrow t^2=\frac{78.4}{4.9} \\ \Rightarrow t^2=16 \\ \Rightarrow t=4second \end{gathered}$$

So, the time taken by the helmet to reach the ground is $4seconds$.

Hence, the time taken $\left(T\right)$ by the sound to reach from the ground to the man is -

$$\begin{gathered} T=t\text{'}-t \\ T=4.23-4 \\ T=0.23seconds \end{gathered}$$

Step 3- To find out the speed of sound in air-

Speed of sound $V=\frac{Dis\tan ce}{time}=\frac{h}{T}$

$$\begin{gathered} \Rightarrow V=\frac{h}{T} \\ \Rightarrow v=\frac{78.4}{0.23}=340.87m/s \end{gathered}$$

Thus, the speed of the sound in air is $340.87m/s.$