A violin player travelling on a slow train sounds a 440 Hz note, while another violin player standing beside the track produces the identical note. As the train approaches the person on the ground when they are near each other, the person on the ground detects 4.0 beats per second. Given the speed of sound in air is 340 m/s , determine the speed of the train.

Step-by-step solution

Let f₀ be the actual frequency of the violin, v be the speed of sound, and v_s be the speed of the train. Given f₀ = 440 Hz and v = 340 m/s . The apparent frequency f' heard by the stationary person on the ground from the approaching train is given by the Doppler effect formula: f' = f₀ ( v v - v_s ) The person hears a beat frequency of 4.0 Hz between the stationary violin and the approaching violin. Thus, the apparent frequency is: f' = f₀ + f_b = 440 + 4 = 444 Hz Substituting the values into the Doppler effect fo