Question

An observer moving with uniform velocity towards a stationary sound source observes frequency f = 170 hz over a distance of x = 80m. If frequency of sound is f₀ = 160Hz and sound travel with speed c = 340 ms⁻¹, then duration of beep emitted by source is n. Find n.

Answer 1

4

Answer 2

The problem involves the Doppler effect for sound. The observer is moving towards a stationary source.

1. Use the Doppler effect formula for an observer moving towards a stationary source to find the observer's velocity $v_o$: $f = f_0 \left( \frac{c + v_o}{c} \right)$ $170 = 160 \left( \frac{340 + v_o}{340} \right)$ $v_o = 21.25 \text{ m/s}$

2. The duration for which the observer hears the beep is the time taken to travel the distance $x = 80 \text{ m}$ with velocity $v_o$: $\Delta t_o = \frac{x}{v_o} = \frac{80}{21.25} = \frac{80}{\frac{85}{4}} = \frac{320}{85} = \frac{64}{17} \text{ seconds}$

3. The total number of waves emitted by the source is equal to the total number of waves received by the observer: $N_{emitted} = f_0 \Delta t_s$ $N_{received} = f \Delta t_o$ $f_0 \Delta t_s = f \Delta t_o$ $\Delta t_s = \Delta t_o \times \frac{f}{f_0}$ $\Delta t_s = \frac{64}{17} \times \frac{170}{160} = \frac{64}{17} \times \frac{17}{16} = \frac{64}{16} = 4 \text{ seconds}$

Therefore, the duration of the beep emitted by the source is $n = 4$ seconds.