Question

Part 1) A cop car traveling at 25 m/s has a siren producing a frequency of 700 Hz. A felon jumps on his motorcycle and speed off in the opposite direction of 15 m/s. What frequency does the felon hear as he sped away (speed of sound is 343 m/s)?

Part 2) The cop does a U-turn and speeds towards the felon at 30 m/s, while the felon speeds up to 20 m/s. What frequency does the felon hear as he sped away (speed of sound is 343 m/s)?
Part 3) What if the felon then sped up to 30 m/s and all other conditions remained the same?

Answer 1

1) 621.8 Hz

2) 719.3 Hz

3) 700 Hz

Step-by-step explanation:

1)

The Doppler effect occurs when there is a source of a wave in relative motion with respect to an observer.

When this happens, the frequency of the wave appears shifted to the observer, according to the equation:

`f'=(v\pm v_o)/(v \pm v_s)f`

where

f is the real frequency of the sound

f' is the apparent frequency of the sound

v is the speed of the sound wave

`v_o` is the velocity of the observer, which is negative if the observer is moving away from the source, positive if the observer is moving towards the source

`v_s` is the velocity of the source, which is negative if the source is moving towards the observer, positive if the source is moving away

In this problem we have:

f = 700 Hz is the frequency of the siren

v = 343 m/s is the speed of sound

`v_s=-25 m/s` is the velocity of the car with the siren

`v_o = +15 m/s` is the velocity of the felon (he's moving away from the siren)

So, the frequency heard by the felon is

`f=(343-25)/(343+15)(700)=621.8 Hz`

2)

In this case, the cop does a U-turn and speeds towards the felon at 30 m/s.

This means that now the siren is moving towards the observer (so,
`v_s` becomes positive), while the sign of
`v_o` still remains positive.

So we have:

f = 700 Hz is the frequency of the siren

v = 343 m/s is the speed of sound

`v_s=+30 m/s` is the velocity of the car with the siren

`v_o = +20 m/s` is the velocity of the felon

So, the frequency heard by the felon is

`f=(343+30)/(343+20)(700)=719.3 Hz`

3)

In this case, the felon speeds up to 30 m/s.

This means that now the felon and the siren are moving with the same relative velocity: so, it's like they are not moving relative to each other, so the frequency will not change.

In fact we have:

f = 700 Hz is the frequency of the siren

v = 343 m/s is the speed of sound

`v_s=+30 m/s` is the velocity of the car with the siren

`v_o = +30 m/s` is the velocity of the felon

So, the frequency heard by the felon is

`f=(343+30)/(343+30)(700)=700 Hz`

So, the frequency will not change.