You blow a whistle of frequency 769 Hz, as you move towards a wall.
The frequency of beats you hear from the whistle and the reflected sound is 11 beats/s.
How fast are you moving towards the wall?
Give your answer in m/s and to two significant figures.
Only answer in numerical values, no units.
To determine how fast you are moving towards the wall, we can use the Doppler effect equation:
f' = (v + v_r) / (v + v_s) * f
Where:
f' is the observed frequency of the sound (which is 769 Hz + 11 Hz)
v is the speed of sound in air (approximately 343 m/s)
f is the emitted frequency of the whistle (769 Hz)
v_r is the velocity of the reflected sound (which we need to find)
v_s is your velocity towards the wall (which we need to find)
Since we are given the observed frequency and the emitted frequency, we can rearrange the equation to solve for the sum of the velocities:
(v + v_r) / (v + v_s) = f' / f
Substituting the given values and rearranging, we have:
(343 + v_r) / (343 + v_s) = (769 + 11) / 769
Simplifying the equation, we have:
(343 + v_r) / (343 + v_s) = 780 / 769
Cross-multiplying, we get:
769 * (343 + v_r) = 780 * (343 + v_s)
Expanding the equation, we have:
263767 + 769v_r = 267240 + 780v_s
Rearranging the equation to isolate v_s, we have:
780v_s = 267240 - 263767 + 769v_r
780v_s = 3473 + 769v_r
v_s = (3473 + 769v_r) / 780
Now, we can substitute the value of v_s in terms of v_r into the original equation. Since the frequency of beats indicates the difference in frequencies between the source and observer, we have:
11 = (v + v_r) / (v + (3473 + 769v_r) / 780) * 769
Simplifying the equation, we have:
11 = (780v + 780v_r) / (780v + 3473 + 769v_r) * 769
Cross-multiplying, we get:
11 * (780v + 3473 + 769v_r) = 769 * (780v + 780v_r)
Expanding the equation, we have:
8580v + 38103 + 8463v_r = 600960v + 600960v_r
Rearranging the equation to isolate v_r, we have:
600960v_r - 8463v_r = 600960v - 8580v - 38103
592497v_r = 592380v - 38103
v_r = (592380v - 38103) / 592497
Now we can substitute this value into our expression for v_s:
v_s = (3473 + 769((592380v - 38103) / 592497)) / 780
To find the speed at which you are moving towards the wall, we need to know the value of v (the speed of sound in air). Without further information, we cannot determine the exact value in numerical terms. Therefore, you need to provide the value of v in order to obtain the numerical value and unit of v_s.