Sonar is used to determine the speed of an object. A 40.0-kHz signal is sent out, and a 42.0-kHz signal is returned. If the speed of sound is 345 m/s, how fast is the object moving?
A) 347 m/s
B) 331 m/s
C) 8.4 m/s
D) 6.9 m/s

Question

Quizplus · Accepted Answer

The Doppler shift in frequency is given by:$$f_d = f_r - f_s = (v \pm v_o) / \lambda$$where:* $f_d$ is the Doppler shift in frequency* $f_r$ is the received frequency* $f_s$ is the source frequency* $v$ is the speed of sound* $v_o$ is the speed of the object* $\lambda$ is the wavelengthIn this case, we have:* $f_r = 42.0 	ext{ kHz}$* $f_s = 40.0 	ext{ kHz}$* $v = 345 	ext{ m/s}$So, the Doppler shift in frequency is:$$f_d = 42.0 	ext{ kHz} - 40.0 	ext{ kHz} = 2.0 	ext{ kHz}$$And the speed of the object is:$$v_o = \lambda f_d = \frac{v f_d}{f_r - f_s} = \frac{(345 	ext{ m/s})(2.0 	ext{ kHz})}{42.0 	ext{ kHz} - 40.0 	ext{ kHz}} = 8.4 	ext{ m/s}$$

Sonar Is Used to Determine the Speed of an Object