Question

If the temperature of air is increased by 5%, then what will be the percentage change in the velocity of sound?

Answer 1

Hint:The speed of sound depends upon the Temperature. The velocity of the sound is given by, $v\, = \,\sqrt {\dfrac{{\gamma RT}}{M}} $. We will use this to find out the percentage change in the velocity of sound.

Complete answer:
We are given that the percentage change in temperature in the question to be 5%
Here only the temperature is increased and the other factors are not changed hence we will only consider the change in temperature.Taking the other factors to be constant:
$v\, = \,\sqrt {\dfrac{{\gamma RT}}{M}} \\
\Rightarrow v = \sqrt {\dfrac{{\gamma R}}{M}} \sqrt T \\
\Rightarrow v = \,k\sqrt T $
Now we have the equation depicting the relation of velocity of sound with temperature.
Now using this the percentage change of velocity of sound: -
\[\dfrac{{\Delta V}}{V}\, \times 100 \\
\Rightarrow \,\dfrac{1}{2}\dfrac{{\Delta T}}{T} \times 100\, \\
\Rightarrow \,\dfrac{1}{2} \times 5\, \\
\Rightarrow \,2.5\% \]

Hence, the percentage change in velocity of sound will be 2.5%.

Additional Information:
Temperature is also a condition that affects the speed of sound. Heat, like sound, is a form of kinetic energy. Molecules at higher temperatures have more energy, thus they can vibrate faster. Since the molecules vibrate faster, sound waves can travel more quickly. The speed of sound in room temperature air is 346 meters per second. This is faster than 331 meters per second, which is the speed of sound in air at freezing temperatures.
The formula to find the speed of sound in air is as follows:
$v = 331m/s + 0.6m/s/C\,\, \times T$
v is the speed of sound and T is the temperature of the air. One thing to keep in mind is that this formula finds the average speed of sound for any given temperature. The speed of sound is also affected by other factors such as humidity and air pressure.

Note:The students should be cautious about the powers when finding the percentage change function from the original function. The students should consider only the parameter which Is changing and given in the question.