An operator seated in his base camp transmits a sound signal of frequency 400 Hz . The signal is reflected back by a car moving towards him. The frequency of the reflected sound is measured to be 410 Hz . Determine the speed of the car, given that the speed of sound in air is 324 m s ⁻¹ .

Step-by-step solution

Let the speed of the car be v_c . The car acts as a moving observer receiving the sound from the stationary base camp. The frequency received by the car is f' = f ( v + v_c v ) . The car then reflects the sound, acting as a moving source. The frequency received by the stationary base camp is f'' = f' ( v v - v_c ) . Combining these, the apparent frequency of the reflected sound is f'' = f ( v + v_c v - v_c ) . Substituting the given values f = 400 Hz , f'' = 410 Hz , and v = 324 m s ⁻¹ : 410 = 400 ( 324 + v_c 324 -