Question

A whistle producing sound waves of frequencies 9500Hz and above is approaching a stationary person with speed v. The velocity of sound in air is 300. If the person can hear frequencies up to a maximum of 10,000Hz, the maximum value of v up to which he can hear whistle is in $m{{s}^{-1}}$
$\begin{align}
  & a)30 \\
 & b)15\sqrt{2} \\
 & c)15 \\
 & d)none \\
\end{align}$

Answer 1

Hint: The whistle sound is heard by a stationary person with some velocity. The source is travelling in this case and the observer is at rest. Next, the maximum frequencies that can be heard by the person is given and also the speed of sound. Using the doppler effect formula, we can solve easily.

Formula used:
$\begin{align}
  & {{v}^{1}}=v(\dfrac{v-{{v}_{0}}}{v-{{v}_{s}}}) \\
 & \\
\end{align}$

Complete step by step answer:
Let us write down the given information,
Velocity of the source=v
Frequency of source is 9500Hz
The speed of observer=0
Maximum Frequency heard by the observer=10,000Hz.
If we observe the given information, the frequency of the source is less than the observer’s maximum frequency meaning that the observer will be able to hear the sound made by the source.
Applying doppler effect formula,
$\begin{align}
  & {{\vartheta }^{1}}=\vartheta (\dfrac{v-{{v}_{0}}}{v-{{v}_{s}}}) \\
 & \Rightarrow 10000=9500(\dfrac{300-0}{300-v}) \\
 & \Rightarrow 100(300-v)=95(300) \\
 & \Rightarrow 300-v=95\times 3 \\
 & \Rightarrow v=300-285 \\
 & \therefore v=15m{{s}^{-1}} \\
 & \\
\end{align}$
Therefore, the correct option is option C.

Additional information:
The doppler effect t is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. This doppler effect is named after the physicist, Christian doppler. He described this phenomenon in 1842. One of the best examples to explain the doppler effect is the horn of a moving vehicle. If a moving vehicle is approaching us, the sound made by the moving vehicle is gradually increasing to us and it will be maximum when we are in the same line with the vehicle. Next, the sound gradually decreases when the vehicle is moving away from us. In this way, we can be moving away from the vehicle or moving towards the vehicle that is at rest and is making sound.

Note:
In the above question, the reason the velocity of the wave is taken positive is because the sound waves made by the moving whistle is moving towards us and is increasing gradually. So, we need to take it as positive. If the same question is told with the vehicle moving away, the velocity of the whistle is taken as negative.