A motorcycle and a police car are moving toward one another. The police car emits sound with a frequency of 512 Hz and has a speed of 28.0 m/s. The motorcycle has a speed of 13.0 m/s.

What frequency does the motorcyclist hear?

To determine the frequency that the motorcyclist hears, we need to apply the concept of the Doppler effect. The Doppler effect describes the change in frequency or pitch of a sound wave as the source of the sound and the observer move relative to each other.

In this scenario, the police car is the source of the sound, emitting sound with a frequency of 512 Hz. The motorcycle is the observer, moving towards the police car.

The formula to calculate the observed frequency (f') due to the Doppler effect when the source is moving and the observer is stationary is given as:

f' = f * (v + v_obs) / (v + v_source),

where:
f = frequency of the source (512 Hz),
v = speed of sound (assumed to be 343 m/s),
v_obs = speed of the observer (13.0 m/s),
v_source = speed of the source (28.0 m/s).

Plugging in the values into the formula, we get:

f' = 512 * (343 + 13) / (343 + 28),

f' = 512 * 356 / 371,

f' ≈ 491.16 Hz.

Therefore, the motorcyclist hears a frequency of approximately 491.16 Hz.