Question

The speed of sound as measured by a student in the laboratory on a winter day is $340m{s}^{-1}$when the room temperature is $17°C$. What speed will be measured by another student repeating the experiment on a day when the room temperature is $32°C$?

Answer 1

Step 1: Given Data:

Speed of sound$\left(V_1\right)=340m{s}^{-1}$

First room temperature $\left(T_1\right)=273+17=290K$

Second room temperature $\left(T_2\right)=273+32=305K$

Step 2: Formula Used:

The speed of sound is given as,

$V=\sqrt{\frac{\gamma RT}{M}}$ where $\gamma =$adiabatic index, $R=$ is the gas constant, $T=$ absolute temperature, and $M=$ Molecular mass.

Here we can see $V\propto \sqrt{T}$

Step 3: Calculation of speed when room temperature changed:

For $\left(T_1\right)=273+17=290K$room temperature,

$V_1\propto \sqrt{T_1}\dots \dots \dots \dots \dots \left(1\right)$

Now the room temperature is, $\left(T_2\right)=273+32=305K$

Let $V_{2}m{s}^{-1}$ be the speed when the room temperature is $305K$

$V_2\propto \sqrt{T_2}\dots \dots \dots \dots \dots \left(2\right)$

Divide equation (1) by equation (2), Substituting the given values

$\begin{array}{rcl}\frac{V_1}{V_2}& =& \sqrt{\frac{T_1}{T_2}}\\ \frac{V_1}{V_2}& =& \sqrt{\frac{T_1}{T_2}}\\ V_2& =& V_1\sqrt{\frac{T_2}{T_1}}\\ & =& 340\sqrt{\frac{305}{290}}\\ & =& 349m{s}^{-1}\end{array}$

Hence, $349m{s}^{-1}$of speed will be measured by another student repeating the experiment on a day when the room temperature is $32°C$.