While standing on 3rd street you hear an approaching ambulance. Its siren oscillates in frequency between about 650 Hz and 750 Hz.
Later on as you near your destination, you come upon the scene of the accident. The now-stationary ambulance runs its siren as it's about to drive off. This time, you hear its pitch oscillate between 637.65 Hz and 735.75 Hz.
Using your physics knowledge, you estimate the speed that the ambulance must have been traveling while en route to the scene of the accident. How fast was it traveling? (Assume it's an average day where the speed of sound is about 343 m/s.)
A. 25 m/s (about 55 mph)
B. 18 m/s (about 40 mph)
C. 13 m/s (about 30 mph)
D. 9 m/s (about 20 mph)
To estimate the speed of the ambulance while en route to the scene of the accident, we need to use the Doppler effect equation for sound:
f' = f * (v + vo) / (v + vs),
Where:
- f' is the observed frequency.
- f is the emitted frequency.
- v is the speed of sound (343 m/s).
- vo is the velocity of the observer (you) relative to the ground.
- vs is the velocity of the source (the ambulance) relative to the ground.
First, we need to find the change in frequency for both cases, when you hear the siren while standing on 3rd street, and when you hear the siren near the accident scene.
When you hear the siren while standing on 3rd street:
Δf = f_max - f_min = 750 Hz - 650 Hz = 100 Hz.
When you hear the siren near the accident scene:
Δf' = f'_max - f'_min = 735.75 Hz - 637.65 Hz = 98.1 Hz.
Next, we can set up the equation using the given information:
f * (v + vo) / (v + vs) = f' * (v + vo) / (v + vs).
We can cancel out (v + vo) on both sides of the equation:
f / (v + vs) = f' / (v + vs).
Now, let's substitute the values we know:
- f = 750 Hz.
- f' = 735.75 Hz.
- v = 343 m/s.
750 / (343 + vs) = 735.75 / (343 + vs).
To solve for vs, let's cross-multiply:
750 * (343 + vs) = 735.75 * (343 + vs).
257250 + 750vs = 252629.1 + 735.75vs.
750vs - 735.75vs = 252629.1 - 257250.
14.25vs = -4620.9.
vs = -4620.9 / 14.25.
vs ≈ -324 m/s.
Since the speed of an ambulance cannot be negative, we take the magnitude of vs:
|vs| ≈ 324 m/s.
The speed of the ambulance must have been approximately 324 m/s.
To convert this speed into miles per hour (mph), we multiply by the conversion factor:
324 m/s * 2.237 = 724.488 mph.
None of the given options match this estimated speed, so it is likely there was an error in the given choices.