Question

The temperature at which the speed of sound in air becomes double of its value at ${27^ \circ }C$?
(A) $ - {123^ \circ }C$
(B) ${927^ \circ }C$
(C) ${327^ \circ }C$
(D) ${54^ \circ }C$

Answer 1

Hint: Temperature is also one of the parameters that affects the sound. If the molecules have higher energy it will vibrate faster. Due to the higher vibration of the molecules the sound will travel more in the air.

Formula used: To solve this type of question we use the following formula.
${v_{air}} = 20.05\sqrt T $ m/s ; this is the speed of sound in air at some temperature T.

Complete step by step answer:
We have to find the temperature at which the speed of sound becomes double of its value at ${27^ \circ }C$.
Let ${v_1}$ is the speed of sound at ${27^ \circ }C$ and ${v_2}$ be the speed at some temperature ${T^ \circ }C$.
Let us use the formula ${v_{air}} = 20.05\sqrt T $ and solve for ${27^ \circ }C = 27 + 273 = 300K$, always use temperature in Kelvin for calculation purposes.
${v_1} = 20.05\sqrt {300} $ (1)
Now similarly use the formula ${v_{air}} = 20.05\sqrt T $ and calculate for temperature T.
${v_2} = 20.05\sqrt T $
Now we know, ${v_2} = 2{v_1}$. Let us substitute this value in the above equation.
$2{v_1} = 20.05\sqrt T $ (2)
Now, substitute equation (1) in equation (2).
$2 \times 20.05\sqrt {300} = 20.05\sqrt T $
On further simplifying we get the following.
$2\sqrt {300} = \sqrt T $
On squaring both the sides we get the following.
$4 \times 300 = T \Rightarrow T = 1200K$
We got the temperature in Kelvin but we need the answer in degree Celsius so let us convert it.
$T = 1200 - 273 = {927^ \circ }C$
Hence, option (B) ${927^ \circ }C$ is the correct option.

Additional information:
Speed of sound in fluid is given as follows.
${v_{fluid}} = \sqrt {\dfrac{{{K_s}}}{\rho }} $ ; Here ${K_s}$ is the coefficient of stiffness.
Speed of sound in Gas is given as follows.
${v_{gas}} = \sqrt {\gamma \dfrac{p}{\rho }} $ ; Here p is the pressure, $\gamma $is adiabatic index and $\rho $ is the density.
Speed of sound in dry air is given as follows.
${v_{dryair}} = 20.05\sqrt T $

Note: The speed of light is variable and depends on the properties of the medium in which it travels.
For calculations, always convert temperature from degree Celsius to Kelvin.
Air molecules have more energies at higher temperatures. That means that the molecules will vibrate faster. This allows the sound waves to travel faster because of the propelled collision between the air molecules.