You stand on the curb waiting to cross the street.Suddenly you hear the sound of a horn from a car approaching you with a constant speed.You hear a frequency of 90 Hz.After the car passes you by,you hear a frequency of 75 Hz.What speed was the car traveling?

Question

Quizplus · Accepted Answer

The Doppler effect causes the frequency of a sound wave to change as the source of the sound moves relative to the observer. The formula for the Doppler effect is:$$f_o = f_s * (v +/- v_o) / (v +/- v_s)$$where:* $$f_o$$ is the observed frequency* $$f_s$$ is the source frequency* $$v$$ is the speed of sound in the medium* $$v_o$$ is the speed of the observer* $$v_s$$ is the speed of the sourceIn this problem, we are given that the observed frequency is 90 Hz when the car is approaching and 75 Hz when the car is receding. We are also given that the speed of sound in air is 343 m/s. We can use the Doppler effect formula to solve for the speed of the car:$$90 Hz = f_s * (343 m/s + 0 m/s) / (343 m/s - v_s)$$$$75 Hz = f_s * (343 m/s - v_s) / (343 m/s + 0 m/s)$$Dividing the first equation by the second equation, we get:$$\frac{90 Hz}{75 Hz} = \frac{343 m/s + 0 m/s}{343 m/s - v_s} * \frac{343 m/s - 0 m/s}{343 m/s + v_s}$$Simplifying, we get:$$\frac{6}{5} = \frac{343 m/s}{343 m/s - v_s} * \frac{343 m/s}{343 m/s + v_s}$$Cross-multiplying, we get:$$6(343 m/s + v_s)(343 m/s - v_s) = 5(343 m/s)^2$$Expanding and simplifying, we get:$$235,848 m^2/s^2 + 686 m/s v_s - 686 m/s v_s - v_s^2 = 117,649 m^2/s^2$$Rearranging, we get:$$v_s^2 = 118,199 m^2/s^2$$Taking the square root, we get:$$v_s = 343 m/s$$Therefore, the speed of the car is 343 m/s, which is approximately 31 m/s.

You Stand on the Curb Waiting to Cross the Street