Question

as you bike toward a loudspeaker, you hear a tone with a frequency of 518 hz. your friend who is biking away from the speaker at the same speed as you hears a tone with a frequency of 482 hz. how fast are each of you traveling? it is a 20∘c day when the speed of sound is 343 m/s.

Answer 1

The student's question involves calculating the speed at which they and their friend are traveling using the Doppler effect. By applying the relevant formulas and given frequencies, it is determined that both are moving at a speed of 12.348 m/s relative to the loudspeaker.

Step-by-step explanation:

The question you're asking involves the phenomenon called the Doppler effect, which is observed when there is a change in frequency of a wave in relation to an observer moving relative to the wave source. To calculate the speed you and your friend are traveling, we would use the Doppler effect equations:

f' = f (v + vr) / (v + vs)

for the observer moving towards the source, and

f' = f (v - vr) / (v - vs)

for the observer moving away from the source, where

• f' is the observed frequency,
• f is the emitted frequency,
• v is the speed of sound,
• vr is the velocity of the receiver, and
• vs is the velocity of the source.

However, since the source is stationary (the loudspeaker), vs is 0. Simplifying the equations and solving for vr (the velocity of the receiver which is you and your friend), we get:
vr = v(f' - f) / (f ± f') (with '+' for the observer moving away and '-' for the observer moving towards the source.)

Using the values you provided:
f' = 518 Hz (you moving toward the speaker), f = 482 Hz (your friend moving away), and v = 343 m/s (speed of sound), we would plug these into the respective equations:

For you moving toward the speaker:
vr = 343 * (518 - 482) / (482 + 518) = 343 * 36 / 1000 = 12.348 m/s, and
vr = 343 * (482 - 518) / (518 - 482) = 343 * -36 / -1000 = 12.348 m/s

Therefore, both you and your friend are traveling at 12.348 m/s.

Answer 2

To calculate the speed at which each person is traveling, we can use the formula for Doppler effect. For the person biking toward the loudspeaker, the speed is 343 m/s. For the friend biking away from the loudspeaker, the speed is also 343 m/s.

Step-by-step explanation:

To calculate the speed at which each person is traveling, we can use the formula for Doppler effect:

f' = f(v + vr)/(v - vs)

Where f is the original frequency, f' is the observed frequency, v is the speed of sound, vr is the speed of the receiver, and vs is the speed of the source.

Let's calculate the speed of each person:

1. For the person biking toward the loudspeaker:
   Given: f = 518 Hz, f' = 518 Hz, v = 343 m/s.
   Substituting the values into the formula:
   518 = 518(343 + vr)/(343 - vs)
   Cancelling out the common factors:
   1 = 343 + vr)/(343 - vs)
   343 - vs = 343 + vr
   vr = vs - 343
   Therefore, the person biking toward the loudspeaker is traveling at a speed of 343 m/s.
2. For the friend biking away from the loudspeaker:
   Given: f = 482 Hz, f' = 482 Hz, v = 343 m/s.
   Substituting the values into the formula:
   482 = 482(343 + vr)/(343 - vs)
   Cancelling out the common factors:
   1 = 343 + vr)/(343 - vs)
   343 - vs = 343 + vr
   vr = -(vs - 343)
   Therefore, the friend biking away from the loudspeaker is also traveling at a speed of 343 m/s.