Question

At what speed should a source of sound move away from a stationary observer so that the observer finds the apparent frequency equal to half the original frequency?

• A. $v/2$
• B. $2v$
• C. $v/4$
• D. $v$

Hint:

Doppler sign conventions: The numerator deals with the observer (moves toward source = $+$, moves away = $-$). The denominator deals with the source (moves toward observer = $-$, moves away = $+$).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This is a classic Doppler Effect problem. A sound source is moving away from a stationary listener. We need to find the source's speed ($v_s$) relative to the speed of sound ($v$) such that the perceived pitch is exactly halved.

Step 2: Key Formula or Approach:
The general Doppler effect formula for frequency is:
$$n' = n \left( \frac{v \pm v_o}{v \mp v_s} \right)$$
Where:
$n'$ = Apparent frequency
$n$ = Original frequency
$v$ = Speed of sound
$v_o$ = Speed of observer (0, since stationary)
$v_s$ = Speed of source
Since the source is moving away, the wavelength stretches, so the denominator must increase. We use the plus ($+$) sign in the denominator.

Step 3: Detailed Explanation:
Given condition: Apparent frequency is half the original.
$$n' = \frac{1}{2} n$$
Substitute this into the specialized Doppler formula for a receding source:
$$\frac{1}{2} n = n \left( \frac{v}{v + v_s} \right)$$
Cancel $n$ from both sides:
$$\frac{1}{2} = \frac{v}{v + v_s}$$
Cross-multiply to solve for $v_s$:
$$v + v_s = 2v$$
Subtract $v$ from both sides:
$$v_s = 2v - v = v$$
The source must move away exactly at the speed of sound (Mach 1).

Step 4: Final Answer:
The source should move at speed $v$, matching option (d).