The Question is from neet-physics waves doppler-effect, and it was created by the author Mr. ANIL KUMAR.G

Question

A bat emitting ultrasonic sound of frequency $f_0$ flies directly towards a large, smooth wall with a velocity $v_s$.  If the speed of sound in air is $v$, and the bat hears the reflected sound, what frequency $f'$ does it perceive?

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
 **Primary Concept**: Doppler Effect
 **Related Topics**: Sound waves, wave reflection
 **Prerequisites**: Understanding of the Doppler effect and how it applies to moving sources and observers.
## SOLUTION PATHWAY
**Given Information**:
  * Frequency emitted by the bat: $f_0$
  * Velocity of the bat (source): $v_s$
  * Speed of sound in air: $v$
  * The bat is moving towards the wall.
  * The wall is considered stationary and acts as a reflector.
**Method & Steps**:
 1. **Frequency received by the wall:** The wall receives a sound wave with a frequency $f_1$ that is higher than $f_0$ due to the Doppler effect. The formula for the Doppler effect when the source is moving towards a stationary observer is:
  $f_1 = f_0 \frac{v}{v - v_s}$
  Reasoning: The bat is moving towards the wall, so the sound waves are compressed, resulting in a higher frequency at the wall.
  Formula/Rule: Doppler effect formula for moving source.
 2. **Frequency reflected by the wall:** The wall reflects the sound wave with the same frequency it received, so the reflected wave has a frequency of $f_1$.
  Working: This is a key step. The reflection doesn't change the frequency.
  Key Point: The wall acts as a new source of sound with frequency $f_1$.
 3. **Frequency perceived by the bat:** The bat is now moving towards the reflected wave (acting as a stationary observer). The frequency perceived by the bat, $f'$, is higher than $f_1$ due to the Doppler effect. The formula is:
  $f' = f_1 \frac{v + v_s}{v}$
  Substituting $f_1$ from step 1:
  $f' = f_0 \frac{v}{v - v_s} \frac{v + v_s}{v} = f_0 \frac{v + v_s}{v - v_s}$
  Result: $f' = f_0 \frac{v + v_s}{v - v_s}$
  Verification: The result makes sense; the frequency perceived by the bat ($f'$) is higher than the emitted frequency ($f_0$) because the bat is moving towards both the emitted and reflected waves.
 ## OPTION ANALYSIS
A ✓: $f' = f_0 \frac{v+v_s}{v-v_s}$: This is the correct formula derived from the Doppler effect calculation for both the outgoing and incoming sound waves.
B :x: $f' = f_0 \frac{v-v_s}{v+v_s}$: This formula incorrectly uses the denominator and numerator in the wrong places, resulting in a lower perceived frequency, which is not physically correct.
C :x: $f' = f_0 \frac{v}{v-v_s}$: This formula only considers the Doppler effect for the outgoing wave and does not account for the bat's movement towards the reflected wave.
D :x: $f' = f_0 \frac{v+v_s}{v}$: This formula only considers the Doppler effect for the incoming wave and doesn't include the change in frequency due to the initial emission towards the wall.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Only considering the Doppler effect for one part of the journey (either emission or reflection):  Remember that the Doppler effect applies twice; once when the bat emits the sound and again when it receives the reflected sound.
  2.Incorrectly applying the Doppler formula: Make sure the correct signs are used in the formula based on the relative motion of the source and observer. If the source and observer are moving towards each other, the denominator will be smaller than the speed of sound.
 **Quick Tips**:
  1.Draw a diagram showing the bat, wall, and sound waves to visualize the situation and relative velocities.
  2.Break the problem into two steps: the emission and the reflection.  Treat the wall as a stationary observer for the emitted wave and a stationary source for the reflected wave.

Mr. ANIL KUMAR.G · Answer

The correct Option is "$f' = f_0 \frac{v+v_s}{v-v_s}$"

Mr. ANIL KUMAR.G · Answer

$f' = f_0 \frac{v-v_s}{v+v_s}$

Mr. ANIL KUMAR.G · Answer

$f' = f_0 \frac{v}{v-v_s}$

Mr. ANIL KUMAR.G · Answer

$f' = f_0 \frac{v+v_s}{v}$

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