Question

The speed of sound in air at ${{0}^{\circ }}$ C is 332 m/s. If it increases at the rate of 0.6 m/s per degree, what will be the temperature when the speed has increased to 344 m/s?

Answer 1

Hint: Assume a variable T which represents the final temperature i.e. temperature at which the speed of sound is 344 m/s. Find the change in speed of sound by subtracting the initial speed and the final speed. It is given that the speed changes by 0.6 m/s with every degree celsius rise in the temperature. So, the temperature T can be found by dividing the change in speed by 0.6.

Complete step-by-step answer:

In this question, we are given that at ${{0}^{\circ }}$ C, the speed of the sound in air is 332 m/s. Also, it is given that this speed increases by 0.6 m/s when there is an increase of $~{{1}^{\circ }}$ C in the temperature. We are required to find the temperature at which the speed of sound is 344 m/s in air.
Let us assume that at temperature T, the speed of sound becomes 344 m/s in air.
Initially, the temperature was ${{0}^{\circ }}$ C. So, the change in temperature is equal to T – 0 = T$^{\circ }$ C.
It is given that for every $~{{1}^{\circ }}$ C rise in the temperature, the speed of sound changes by 0.6 m/s. So, for T$^{\circ }$ C rise in the temperature, the speed of the sound increases by 0.6T m/s . . . . . . . . . . (1)
It is given that the initial speed of the sound is 332 m/s and the final speed of the sound is 344 m/s. So, the change in the speed of sound is also equal to 344 - 332 = 12 m/s . . . . . . . . (2)
From equation (1) and equation (2), we can say,
0.6 T = 12
$\begin{aligned} & \Rightarrow \text{T = }\dfrac{12}{0.6} \\\ & \Rightarrow \text{T = 2}{{\text{0}}^{\circ }}C \\\ \end{aligned}$
Hence, the temperature at which the speed of the sound is 344 m/s is $\text{2}{{\text{0}}^{\circ }}C$.

Note: We can also do this question by dimensional analysis if we do not know how to do this question mathematically. If we observe the dimensions of all the numbers given in the question, we can solve this question with the help of dimensions only. For example, it is given that the speed increases by 0.6 m/s per degree, this means that for every degree rise in the temperature, the speed of the sound increases by 0.6 m/s.