Question

You are falling down a pit that is 2400m deep. As you fall, You hear a pitch from the surface that is 1000 Hz.

a. Find the min and max frequencey you hear as you fall.

b. Find the half-max frequeny that you hear, at what depth are you at?

Answer 1

a) At the top of the pit, my velocity is zero. So, observed frequency is 1000 Hz

At the bottom of the pit, my velocity is,

v = (2gh)^1/2 = (2 * 9.81 * 2400)^1/2 = 217 m/s

Due to Doppler effect, observed frequency at the bottom of the pit is,

f_bottom = [(340 - 217) / 340] * 1000 = 362 Hz

So, 1000 Hz is the maximum frequency and 362 Hz is the minimum freqiency.

b) Half max frequency = 500 Hz

Let the depth at which I hear the half frequency be 'd'.

Velocity at depth 'd' is, v_d = (2gd)^1/2

500 = [(340 - v_d) / 340] * 1000

=> 340 - v_d = 170

=> v_d = 170

=> (2gd)^1/2 = 170

=> d = 170² / (2 * 9.81) = 1473 m