Question

A construction worker’s hamlet slips and falls when he is $78.4\;{\text{m}}$ above the ground. He hears the sound of the helmet hitting the ground $4.23\;{\text{seconds}}$ after it slipped. Find the speed of the sound in the air.
A $340.87\;{\text{m}}/{\text{s}}$
B $350.2\;{\text{m}}/{\text{s}}$
C $349.63\;{\text{m}}/{\text{s}}$
D $186.3\;{\text{m}}/{\text{s}}$
E None of these

Answer 1

The above problem can be solved by the equation of motion for the object under gravity. The speed of the sound can be found by calculating the time in which the sound travels to the ground.

Complete step by step solution:
Given: The height of the construction worker above the ground is $h = 78.4\;{\text{m}}$, the time for hearing the sound is $T = 4.23\;{\text{s}}$.
The equation to calculate the time taken by the helmet to react the ground is given as:
$h = ut + \dfrac{1}{2}g{t^2}......\left( 1 \right)$
Here, u is the initial speed of the helmet and its value is 0, g is the gravitational acceleration and its value is $9.8\;{\text{m}}/{{\text{s}}^2}$.
Substitute $78.4\;{\text{m}}$for h, 0 for u and $9.8\;{\text{m}}/{{\text{s}}^2}$for g in the equation (1) to find the time in which the helmet reaches to ground.
$78.4\;{\text{m}} = \left( 0 \right)t + \dfrac{1}{2}\left( {9.8\;{\text{m}}/{{\text{s}}^2}} \right){t^2}$
$78.4\;{\text{m}} = \dfrac{1}{2}\left( {9.8\;{\text{m}}/{{\text{s}}^2}} \right){t^2}$
${t^2} = 16\;{{\text{s}}^2}$
$t = 4\;{\text{s}}$
The equation to hear the sound after reaching the helmet to ground is given as:
$\Rightarrow$ ${t_1} = T - t......\left( 2 \right)$
Substitute $4.23\;{\text{s}}$ for T and $4\;{\text{s}}$ for t in the equation (2) to find the time in which the worker hears the sound.
$\Rightarrow$ ${t_1} = 4.23\;{\text{s}} - 4\;{\text{s}}$
${t_1} = 0.23\;{\text{s}}$
The equation to find the speed of the sound is given as:
$v = \dfrac{h}{{{t_1}}}......\left( 3 \right)$
Substitute $78.4\;{\text{m}}$for h and $0.23\;{\text{s}}$ for ${t_1}$ in the equation (3) to find the speed of the sound.
$\Rightarrow$ $v = \dfrac{{78.4\;{\text{m}}}}{{0.23\;{\text{s}}}}$
$\therefore$ $v = 340.87\;{\text{m}}/{\text{s}}$

Thus, the speed of the sound is $340.87\;{\text{m}}/{\text{s}}$ and the option (A) is correct.

Note: Use the value of gravitational acceleration as $9.8\;{\text{m}}/{{\text{s}}^2}$ because the approximate value of gravitational acceleration does not provide the correct answer from the given options. Use the significant digits in the calculated speed of the sound to match the correct option.