Question

At rest, a car's horn sounds the note A (440 Hz). The horn is sounded while the car is moving down the street. A bicyclist moving in the same direction with one-third the car's speed hears a frequency of 403 Hz.

(a) Is the cyclist ahead of or behind the car?

ahead/behind    

(b) What is the speed of the car?
(Answer in m/s)

Answer 1

1. Since the frequency of the sound that is heard by the bicyclist is lower in frequency than the original sound, we know that the car must be moving away from the bicyclist, i.e. the bicyclist is behind the car, moving at a speed that is 1/3 that of the cars. Behind the car, we know that the wavelength of the sound is given by

   l' = l + v_s/f

   This corresponds to a frequency that is given by

   f' = v/l' = f v/(v + v_s)

   This is the frequency relative to still air. As the bicyclist moves through this wave, she hears a frequency that is increased above this still air frequency by

   f'' = f'(v + v_o)/v = f (v + v_o)/(v + v_s)

   If we use the fact that 3v_o = v_s, we get

   f'' = f (v + v_o)/(v + 3v_o)

   Solving this for v_o, we get

   f'' (v + 3v_o) = f (v + v_o)
   v_o (3f'' - f) = v (f - f'')
   v_o = v (f - f'')/(3f'' - f) = 345 m/s (25/805) = 10.7 m/s

   Since the car is moving 3 times faster, the speed of the car is 32.1 m/s