# Doppler effect – problems and solutions

1.

(1) an observer moving toward the stationery source

(2) source moving toward the stationary observer

(3) observer and source approach each other

(4) observer and source are moving at the same speed

If the pitch heard is higher than that of the emitted source frequency, then which statement above are correct :

A. (1), (2) and (3)

B. (1), (2), (3) and (4)

C. (1) and (3)

D. (1) and (4)

E. (2) and (4)

Solution

The equation of the Doppler effect :

Sign rule :

The sound speed (v) always positive

The observer speed (v_{obs}) is positive if observer moving toward the source of the sound

The observer speed (v_{obs}) is negative if the observer moving away from the source of the sound

The source speed (v_{source}) is positive if the source of the sound moving away from the observer

The source speed (v_{source}) is negative if the source of the sound moving toward the observer

The observer speed (v_{obs}) = 0 if an observer at rest

The source speed (v_{source}) = 0 if source at rest

For example :

The observer speed (v_{obs}) = 60 m/s, if observer at rest then v_{obs }= 0

The source speed (v_{source}) = 40 m/s, if the source of the sound at rest then v_{source} = 0

The sound speed (v) = 340 m/s

The frequency of sound (f) = 1000 Hertz

__An observer moving toward the stationery source__

*The o**bserver **speed **(**v*_{obs}*) is** positive if **observer moving toward **the source of the sound*

*The source of the sound at rest so **v*_{source }*= 0*

__Source moving toward the stationary observer__

*The source speed (v _{source}) is negative if the source of the sound moving toward an observer*

*Observer at rest so **(**v*_{obs}*) **= 0*

__Observer and source approach each other__

*The observer speed (v _{obs}) is positive if observer moving toward the source of the sound*

*The source speed (v*_{source}*) is negative if the source of the sound moving toward the observer*

__Observer and source are moving at the same speed__

If the source of the sound and observer moves at the same speed then no Doppler effect occurs.

2. An observer at rest near the source of the sound of frequency 684 Hz. Another the source of the sound of 676 Hz moving toward the observer at 2 n/s. If the speed of the sound waves in air is 340 m/s, then what is the beat frequency heard by the observer.

__Known :__

The frequency of the source of the sound 1 (f_{1}) = 684 Hz (rest)

The frequency of the source of the sound 2 (f_{2}) = 676 Hz (move)

The speed of the source of the sound 2 (v_{2}) = 2 m/s (moving toward the observer)

The speed of the source of the sound waves in air (v) = 340 m/s

__Wanted:__ The beat frequency heard by the observer

__Solution :__

The equation of the Doppler effect :

Sign rule :

The sound speed (v) always positive

The observer speed (v_{obs}) is positive if observer moving toward the source of the sound

The observer speed (v_{obs}) is negative if observer moving away from the source of the sound

The source speed (v_{source}) is positive if the source of the sound moving away from the observer

The source speed (v_{source}) is negative if the source of the sound moving toward the observer

The observer speed (v_{obs}) = 0 if an observer at rest

The source speed (v_{source}) = 0 if source at rest

The beat frequency heard by the observer = 684 Hz – 680 Hz = 4 Hz.

3. A source of sound moving toward the stationary observer at 20 m/s. The frequency of the source of the sound = 380 Hz. The speed of the sound waves in air = 400 m s^{-1}. What is the frequency of the sound waves heard by the observer?

__Known :__

The speed of the source of the sound (v_{source}) = 20 m/s

The speed of observer (v_{p}) = 0

The frequency of the source of the sound (f) = 380 Hz

The speed of the source of the sound waves (v) = 400 m s^{-1}

__Wanted:__ The frequency of the sound waves heard by the observer

4. Car A moves at 72 km/h and car B moves at 90 km/h, approach each other. Car A honked with a frequency of 650 Hz. If the speed of the sound waves in air is 350 m/s, then what is the frequency of sound heard by the driver of car B from car A.

__Known :__

The speed of car A (v_{A}) = 72 km/h = 20 m/s, approach car B

The speed of car B (v_{B}) = 90 km/h = 25 m/s, approach car A

The frequency of the sound of car A (f_{A}) = 650 Hz

The speed of the sound waves in air (v) = 350 m/s

Wanted: Frequency of sound heard by the driver of the car B from car A

__Solution :__

The sound speed (v) always positive

The observer speed (v_{obs}) is positive if observer moving toward the source of the sound

The observer speed (v_{obs}) is negative if the observer moving away from the source of the sound

The source speed (v_{source}) is positive if the source of the sound moving away from the observer

The source speed (v_{source}) is negative if the source of the sound moving toward the observer

The observer speed (v_{obs}) = 0 if an observer at rest

The source speed (v_{source}) = 0 if source at rest

5. A source of sound moves at 10 m/s approach a stationary observer. The frequency of the source of the sound is 380 Hz and the speed of the sound waves in air is 400 m/s What is the frequency of the sound waves heard by the observer.

__Known :__

The speed of the source of the sound (v_{s}) = 20 m/s

The speed of observer (v_{p}) = 0

The frequency of the source of the sound (f) = 380 Hz

The speed of the sound waves in air (v) = 400 m/s

__Wanted:__ The frequency of the sound waves heard by the observer

__Solution :__

The sound speed (v) always positive

The observer speed (v_{obs}) is positive if observer moving toward the source of the sound

The observer speed (v_{obs}) is negative if the observer moving away from the source of the sound

The source speed (v_{source}) is positive if the source of the sound moving away from observer

The source speed (v_{source}) is negative if the source of the sound moving toward the observer

The observer speed (v_{obs}) = 0 if an observer at rest

The source speed (v_{source}) = 0 if source at rest

6. A car moving toward a stationary observer that emits 490 Hz sound wave. The beat frequency heard is 10 Hz. If the speed of the sound waves in air is 340 m/s, what is the speed of the car?

__Known :__

The frequency of sound (f) = 490 Hertz

The speed of the sound waves in air (v) = 340 m/s

The observer approaches the sound source so that the frequency of sound heard is greater than the frequency of the sound source. The frequency of sound = 490 Hertz and the beat frequency = 10 Hertz so that the frequency of the sound heard by an observer (f’) = 500 Hertz.

__Wanted:__ the speed of the car

__Solution :__

The equation of the Doppler effect :

Sign rule :

The sound speed (v) always positive

The observer speed (v_{obs}) is positive if observer moving toward the source of the sound

The observer speed (v_{obs}) is negative if the observer moving away from the source of the sound

The source speed (v_{source}) is positive if the source of the sound moving away from the observer

The source speed (v_{source}) is negative if the source of the sound moving toward the observer

The observer speed (v_{obs}) = 0 if an observer at rest

The source speed (v_{source}) = 0 if source at rest

The speed of the car is 6.9 m/s.

7. The police car that was ringing a 930 Hz siren chased after someone who ran away on a motorcycle with a speed of 72 km.jam^{-1}. The speed of police cars reaches 108 km.hour^{-1}. If the speed of sound in the air is 340 m.s-1, then the frequency of siren sounds heard by motorcyclists is …

Solution :

Sign rule :

The sound speed (v) always positive

The observer speed (v_{obs}) is positive if observer moving toward the source of the sound

The observer speed (v_{obs}) is negative if the observer moving away from the source of the sound

The source speed (v_{source}) is positive if the source of the sound moving away from the observer

The source speed (v_{source}) is negative if the source of the sound moving toward the observer

The observer speed (v_{obs}) = 0 if an observer at rest

The source speed (v_{source}) = 0 if source at rest

__Known :__

The frequency of the source of sound (f) = 930 Hz

The speed of observer (v_{p}) = 72 km.hour^{-1} = 72 (1000 meters) / 3600 (seconds) = 72,000/3600 meters/second = 20 m/s = -20 m.s^{-1}

The speed of the source of sound (v_{source}) = 108 km.hour^{-1 }= 108 (1000 meters) / (3600 seconds) = 108,000 / 3600 meters/second = 30 m/s = -30 m.s^{-1}

The speed of the source of the sound waves (v) = 340 m.s^{-1}

__Wanted :__ The frequency of the sound waves heard by an observer (f’)

__Solution :__

The equation of the Doppler effect :

8. An ambulance moves at 72 km.hour^{-1} while sounding a siren with a frequency of 1500 Hz. Motorcyclists move at speeds of 20 m.s^{-1} in opposite directions with ambulances. If the speed of sound in air is 340 m.s^{-1}, then the ratio of frequencies heard by motorcyclists when approaching and away from the ambulance is …

__Known :__

The frequency of the source of sound (f) = 1500 Hz

The speed of observer (v_{p}) = 20 m/s

The speed of the source of sound (v_{source}) = 72 km/hour = 72 (1000 meters) / 3600 seconds = 72,000/3600 meters/seconds = 20 m/s

The speed of the source of the sound waves (v) = 340 m/s

__Wanted:__ The ratio of frequencies heard by motorcyclists when approaching and away from the ambulance

__Solution :__

The equation of the Doppler effect :

**Frequencies heard by motorcyclists approaching ambulances**

*Both are in opposite directions so that when the motorcycle approaches the ambulance car, the two approach each other. v*_{p }*is positive if the listener approaches the sound source and v*_{s }*is **negative if the sound source approaches the listener.*

**Frequencies are heard by motorcyclists as they move away from the ambulance**

*Both are opposite direction so that when the motorcycle away from the ambulance, both of them away from each other. v _{p }is negative if the listener is away from the sound source and v_{s} is positive if the sound source is away from the listener.*

**T****he ratio of frequencies heard by motorcyclists when approaching and away from the ambulance**