Page 201 - Schaum's Outline of Theory and Problems of Applied Physics

186 WAVES AND SOUND [CHAP. 15

8
Here c is the velocity of light (3.00 × 10 m/s), and v is the relative velocity between source and observer
(considered positive if they are approaching and negative if they are receding). Astronomers use the Doppler
effect in light to determine the motions of stars. Police use the effect in “radar guns” that send out pulses of
short-wavelength radio waves that are reflected from a moving vehicle. When the reflected pulses return to the
radar gun, the Doppler shift in their frequency is measured and reveals the speed of the vehicle.

SOLVED PROBLEM 15.14
The siren of a fire engine has a frequency of 500 Hz. (a) The fire engine approaches a stationary car at
20 m/s. What frequency does a person in the car hear? (b) The fire engine stops and the car drives away
from it at 20 m/s. What frequency does the person in the car hear now?

(a) Here f S = 500 Hz, v = 343 m/s, v S =+20 m/s, and v L = 0. Hence the perceived frequency of the siren is

v + v L 343 m/s
f L = f S = (500 Hz) = 531 Hz
v − v S (343 − 20) m/s
(b) In this case v s = 0 and v L =−20 m/s. The perceived frequency of the siren is

v + v L (343 − 20) m/s
f L = f S = (500 Hz) = 471 Hz
v − v S 343 m/s
SOLVED PROBLEM 15.15
A person arriving late at a concert hurries toward her seat so fast that the note middle C (262 Hz) appears
1 Hz higher in frequency to her. How fast is she moving?
The first step is to solve for v L the formula for the frequency the listener hears. Since v S = 0,

v + v L
f = f S
v
f v + v L v L
= = 1 +
f S v v
v L f
= − 1
v f S
f

v L = v − 1
f S
Here v =+343 m/s, f = 262 Hz, and f L = 263 Hz, so

263 Hz
v L = (343 m/s) − 1 = 1.3 m/s
262 Hz
SOLVED PROBLEM 15.16
7
A distant galaxy of stars in the constellation Hydra is moving away from the earth at 6.1 × 10 m/s. One
of the characteristic wavelengths in the light the galaxy emits is 5.5×10 −7 m. What is the corresponding
wavelength measured by astronomers on the earth?

Since f = c/λ and f S = c/λ S ,wehave
1/2

1 + (v/c)
f = f S
1 − (v/c)
1/2

c c 1 + (v/c)
=
λ λ S 1 − (v/c)
1/2

1 − (v/c)
λ = λ S
1 + (v/c)