Two motorcycles are traveling in opposite directions at the same speed when one of the
cyclists blasts his horn, which has frequency of 544 Hz. The other cyclist hears the
frequency as 522 Hz. If the speed of sound in air is 344 m/s, what is the speed of the
motorcycles
The motorcyclists are going in opposite directions, away from one another. Each has speed v and the sound speed is a. The received frequency is
522 = 544*(a-v)/(a+v)
(344-v)/(344+v) = 0.95956
344-v = 330.09 + 0.94956v
1.94956v = 13.912
v = 7.1 m/s
To solve this problem, we can use the Doppler effect formula for sound.
The formula is: f = (v + vo)/(v + vs) * fo
Where:
f = observed frequency
fo = actual frequency (in this case, 544 Hz)
v = speed of sound in air (344 m/s)
vo = velocity of the observer (one of the cyclists)
vs = velocity of the source (the other cyclist)
The observed frequency is 522 Hz, so we can substitute the values into the formula:
522 = (344 + vo) / (344 + vs) * 544
Now we can rearrange the equation to solve for the velocities:
522 / 544 = (344 + vo) / (344 + vs)
0.959559 = (344 + vo) / (344 + vs)
Cross-multiplying:
0.959559 * (344 + vs) = 344 + vo
329.959 = 344 + vo - 0.959559vs
Now, let's subtract 344 from both sides:
329.959 - 344 = vo - 0.959559vs
-14.041 = vo - 0.959559vs
Finally, let's rearrange the equation to solve for the difference in velocities:
0.959559vs = vo + 14.041
Divide both sides by 0.959559:
vs = (vo + 14.041) / 0.959559
Now we have the equation to find the difference in velocities between the two motorcycles. However, since they are traveling in opposite directions at the same speed, their velocities should have the same magnitude, but with opposite signs. Therefore, we can rewrite the equation as:
vs = -(vo + 14.041) / 0.959559
This equation gives us the velocity of one motorcycle relative to the other.
To solve this problem, we can use the Doppler effect formula, which relates the observed frequency to the source frequency and the relative velocity.
The formula for the Doppler effect is as follows:
(observed frequency) = (source frequency) * (speed of sound + velocity of observer) / (speed of sound + velocity of source)
Let's denote the speed of sound as V, the velocity of the observer as Vobserver, and the velocity of the source as Vsource. We are given the following information:
Source frequency (fsource) = 544 Hz
Observed frequency (fobserved) = 522 Hz
Speed of sound (V) = 344 m/s
Using the Doppler effect formula and applying it to our problem, we have:
fobserved = fsource * (V + Vobserver) / (V + Vsource)
Now we can substitute the given values and solve for Vsource:
522 Hz = 544 Hz * (344 m/s + Vobserver) / (344 m/s + Vsource)
To simplify the calculation, let's assume the speed of the observer (Vobserver) is zero since it is not given that the observer is in motion relative to the air. Therefore, the equation becomes:
522 Hz = 544 Hz * 344 m/s / (344 m/s + Vsource)
Simplifying further:
522 Hz * (344 m/s + Vsource) = 544 Hz * 344 m/s
Now, let's solve for Vsource:
522 * 344 + 522 * Vsource = 544 * 344
179,568 + 522Vsource = 187,616
522Vsource = 8,048
Vsource = 8,048 / 522
Vsource ≈ 15.404 m/s
Therefore, the speed of the motorcycles is approximately 15.404 m/s.