Question

When an observer moves towards a stationary source with velocity '\(V_1\)', the apparent frequency of emitted note is '\(F_1\)'. When observer moves away from stationary source with velocity '\(V_1\)' the apparent frequency is '\(F_2\)'. If '\(v\)' is velocity of sound in air and \(\frac{F_1}{F_2} = 2\), then \(\frac{v}{V_1}\) is equal to

• A. 6
• B. 5
• C. 3
• D. 4

Hint:

Shortcut: If $F_1/F_2 = n$, then $v/V_1 = (n+1)/(n-1)$. Here $(2+1)/(2-1) = 3$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem applies the Doppler effect for sound waves.
The apparent frequency heard by an observer changes depending on the relative motion between the observer and the source of the sound.
Step 2: Key Formula or Approach:
The general formula for apparent frequency \(f'\) due to the Doppler effect is:
\[ f' = f_0 \left( \frac{v \pm v_o}{v \mp v_s} \right) \]
where \(f_0\) is the true frequency emitted, \(v\) is the speed of sound, \(v_o\) is the observer's velocity, and \(v_s\) is the source's velocity.
Step 3: Detailed Explanation:
In the given problem, the source is stationary, which means \(v_s = 0\).
Case 1: The observer moves towards the source with velocity \(V_1\).
Approaching the source increases the apparent frequency, so we use the plus sign (\(+\)) in the numerator.
\[ F_1 = f_0 \left( \frac{v + V_1}{v} \right) \]
Case 2: The observer moves away from the source with velocity \(V_1\).
Receding from the source decreases the apparent frequency, so we use the minus sign (\(-\)) in the numerator.
\[ F_2 = f_0 \left( \frac{v - V_1}{v} \right) \]
We are given the condition that the ratio of these frequencies is:
\[ \frac{F_1}{F_2} = 2 \]
Substituting our expressions for \(F_1\) and \(F_2\) into this ratio:
\[ \frac{f_0 \left( \frac{v + V_1}{v} \right)}{f_0 \left( \frac{v - V_1}{v} \right)} = 2 \]
The terms \(f_0\) and the denominator \(v\) cancel out:
\[ \frac{v + V_1}{v - V_1} = 2 \]
Now, solve for the relationship between \(v\) and \(V_1\). Cross-multiply:
\[ v + V_1 = 2(v - V_1) \]
\[ v + V_1 = 2v - 2V_1 \]
Rearrange the terms to group \(v\) and \(V_1\):
\[ V_1 + 2V_1 = 2v - v \]
\[ 3V_1 = v \]
Finally, we want the ratio \(\frac{v}{V_1}\):
\[ \frac{v}{V_1} = 3 \]
Step 4: Final Answer:
The value of the ratio is \(3\).