Worksheet: Frequency Changes due to the Doppler Effect

In this worksheet, we will practice determining the frequencies of sound waves as measured by observers moving relative to the sources of the sounds.

Q1:

A wave is emitted from a source that moves relative to a stationary observer at a velocity 𝑣  , where a positive value of 𝑣  corresponds to the motion of the source toward the observer. The wave propagates with a velocity 𝑣 . Which of the following formulas correctly relates the frequency 𝑓 , measured at the wave’s source, and the frequency 𝑓 β€² of the wave measured by the observer?

  • A 𝑓 β€² = 𝑓 ( 𝑣 βˆ’ 𝑣 ) ( 𝑣 + 𝑣 )  
  • B 𝑓 β€² = 𝑓 𝑣 ( 𝑣 βˆ’ 𝑣 ) 
  • C 𝑓 β€² = 𝑓 ( 𝑣 + 𝑣 ) ( 𝑣 βˆ’ 𝑣 )  
  • D 𝑓 β€² = 𝑓 𝑣 ( 𝑣 + 𝑣 ) 

Q2:

A fire alarm activated on one floor of a building where a fire drill is occurring emits sound waves that propagate at 343 m/s. A person leaving the building accidentally drops their phone down a stairwell during a conversation, and the listener hears the fire alarm at the instant the phone hits the ground as having a frequency of 12.3 kHz. The phone is dropped from the floor of the building where the alarm is active, and it hits the ground traveling at 18 m/s. What is the frequency of the alarm? Give your answer to three significant figures.

Q3:

A wave is emitted from a stationary source and propagates with a velocity 𝑣 . Which of the following formulas correctly relates the frequency 𝑓 , measured at the wave’s source, and the frequency 𝑓  of the wave measured by an observer moving relative to the source at a velocity 𝑣  , where a positive value of 𝑣  corresponds to the motion of the observer toward the source?

  • A 𝑓 = 𝑓 ( 𝑣 + 𝑣 ) ( 𝑣 βˆ’ 𝑣 )   
  • B 𝑓 = 𝑓 ( 𝑣 + 𝑣 ) 𝑣  
  • C 𝑓 = 𝑓 ( 𝑣 βˆ’ 𝑣 ) ( 𝑣 + 𝑣 )   
  • D 𝑓 = 𝑓 ( 𝑣 βˆ’ 𝑣 ) 𝑣  

Q4:

Sound waves with a frequency of 1.86 kHz are emitted by a man shouting to a passing cyclist. The cyclist is moving at βˆ’ 1 1 m/s relative to the man. The sound waves propagate at 343 m/s. What is the frequency of the sound waves from the man perceived by the cyclist? Give your answer to three significant figures.

Q5:

Sound waves propagating at 343 m/s are emitted from a vibrating jackhammer used on a maintenance site at the side of a road. A car on the road is moving toward the jackhammer at 24 m/s. The frequency of the sound waves from the jackhammer perceived by the driver of the car is 972 Hz. What is the frequency of vibration of the jackhammer? Give your answer to three significant figures.

Q6:

An airport worker listens to an airliner landing, with its engines emitting sound waves with a frequency of 444 Hz as it approaches the worker at a velocity of 45.5 m/s. The sound waves propagate at 341 m/s.

What frequency of sound is perceived by the worker to be emitted from the airliner? Answer to three significant figures.

What frequency of sound is perceived by another worker to be emitted from the airliner, assuming that the other worker is on the opposite side of the airliner to the first worker and the airliner moves along a line that intersects both workers? Answer to three significant figures.

Q7:

A traveler on a train station platform hears a train approaching. The train produces a sound of frequency 7.44 kHz, which the traveler perceives as having a frequency of 6.98 kHz. The sound waves from the train propagate at 344 m/s. At what speed is the train approaching the platform?

Q8:

A bungee jumper jumps from a bridge and then begins to rise up toward it due to the contraction of the bungee cord. The jumper perceives the cheering of some onlookers on the bridge as having a frequency of 3.21 kHz at an instant 𝑑 . The cheering onlookers perceive the frequency of their cheers as 3.19 kHz. The sound of the cheering propagates at 343 m/s. What is the speed at which the bungee jumper is moving at the instant 𝑑 ?

Q9:

Sound waves with a frequency 2.25 kHz are emitted from a loudspeaker used in a public-address system at a motor sports event. A car participating in a race is moving toward the speaker at 33 m/s. The sound waves propagate at 343 m/s. What is the frequency of the sound waves from the speaker perceived by the driver of the car? Answer to three significant figures.

Q10:

Two aircraft are traveling in the same direction as each other, with the leading aircraft directly ahead of the following aircraft, as shown in the diagram. The diagram is not drawn to scale. The leading aircraft is traveling at 288 m/s. The frequency of the sound waves from the following aircraft as measured by the leading aircraft is 1.11 times the frequency of the sound waves of the following aircraft as measured by the following aircraft itself. Sound waves in the air between the two aircraft propagate at 350 m/s. What is the speed of the following aircraft?

Q11:

A child on a roller coaster in an amusement park passes another joyride. The other ride is stationary and is emitting a musical note with a frequency of 1,024 Hz. The child perceives the note’s frequency as 1,002 Hz. The sound waves propagate at 343 m/s. At what speed does the roller coaster move away from the other ride? Give your answer to three significant figures.

Q12:

Two objects that emit 425 Hz sounds move directly away from each other, both moving at 72.5 m/s. The ratio of the frequencies of the sound waves of each object from itself and from the other object is 1.55. What speed do the sound waves propagate at in the air around the objects?

Q13:

Two aircraft travel in the same direction, with the leading aircraft directly ahead of the following aircraft, as shown in the diagram. The diagram is not drawn to scale. The leading aircraft is traveling at 296 m/s. The frequency of the sound waves from the following aircraft as measured by the leading aircraft is 0.750 times the frequency of the sound waves of the following aircraft as measured by the following aircraft itself. Sound waves in the air between the aircraft propagate at 350 m/s. What is the speed of the following aircraft?

Q14:

Two jet fighters, jet A and jet B, are flying directly away from each other, as shown in the diagram. The diagram is not drawn to scale. Each of them emits a sound with a frequency of 4.42 kHz as measured by the pilots of the jets. Jet A travels at 233 m/s and jet B travels at 256 m/s. Sound waves move in the air around the jets at 349 m/s.

At what frequency does the pilot of jet A hear the sounds from jet B?

At what frequency does the pilot of jet B hear the sounds from jet A?

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