To solve this problem, we need to analyze the situation using concepts from physics, particularly those related to motion and the Doppler effect. The motorcycle accelerates from rest, and we want to find out how far it travels when the frequency of the sound from the siren is perceived at 94% of its original value. Let's break this down step by step.
Understanding the Doppler Effect
The Doppler effect describes how the frequency of a wave changes for an observer moving relative to the source of the wave. In this case, the motorcycle is moving away from a stationary siren. The formula for the observed frequency (f') when the source is stationary and the observer is moving away is:
f' = f (v / (v + v_o))
Where:
- f' = observed frequency
- f = source frequency (original frequency of the siren)
- v = speed of sound in air (330 m/s)
- v_o = speed of the observer (motorcycle)
Setting Up the Problem
We know that the observed frequency is 94% of the original frequency, so we can express this as:
f' = 0.94f
Substituting this into the Doppler effect formula gives us:
0.94f = f (v / (v + v_o))
We can simplify this equation by canceling out the frequency (f) from both sides (assuming f is not zero):
0.94 = v / (v + v_o)
Finding the Speed of the Motorcycle
Rearranging the equation to solve for the speed of the motorcycle (v_o) gives us:
0.94(v + v_o) = v
Expanding this leads to:
0.94v + 0.94v_o = v
Now, isolating v_o:
0.94v_o = v - 0.94v
0.94v_o = 0.06v
v_o = (0.06v) / 0.94
Substituting the speed of sound (v = 330 m/s):
v_o = (0.06 * 330) / 0.94 ≈ 21.1 m/s
Calculating the Distance Traveled
Now that we have the speed of the motorcycle when the frequency is perceived at 94% of its value, we can find the distance it has traveled. The motorcycle starts from rest and accelerates at a rate of 2 m/s². We can use the kinematic equation:
s = ut + (1/2)at²
Where:
- s = distance traveled
- u = initial velocity (0 m/s, since it starts from rest)
- a = acceleration (2 m/s²)
- t = time taken to reach the speed v_o
First, we need to find the time (t) it takes to reach the speed of 21.1 m/s using the formula:
v_o = u + at
Substituting the known values:
21.1 = 0 + 2t
Solving for t gives:
t = 21.1 / 2 ≈ 10.55 seconds
Now we can substitute t back into the distance equation:
s = 0 * 10.55 + (1/2) * 2 * (10.55)²
s = (1) * (111.3025) ≈ 111.3 meters
Final Result
The motorcycle has traveled approximately 111.3 meters when the driver hears the frequency of the siren at 94% of its original value. This example illustrates the interplay between motion and sound, showcasing how relative speeds affect our perception of frequency.
