A speedboat, starting from rest, moves along a straight line away from the dock. The boat has a constant acceleration of +3.00 m/s2. A siren on the dock is producing a 755 Hz tone. Assuming that the air temperature is 20.0oC, what is the frequency of the sound heard by a person on the boat when the boat’s displacement from the dock is +45.0 m?
The frequency of the sound heard by a person on the boat when the boat's displacement from the dock is +45.0 m is 745.7 Hz. This is due to the Doppler effect, which states that the frequency of a sound wave is shifted when the source and observer are in motion relative to each other. The frequency of the sound wave is shifted down by an amount proportional to the relative velocity between the source and observer. In this case, the relative velocity is the speed of the boat, which can be calculated using the equation v = at, where v is the velocity, a is the acceleration, and t is the time. Since the boat is starting from rest, the time is equal to the displacement divided by the acceleration, or t = 45.0 m/ 3.00 m/s2 = 15.0 s. Therefore, the speed of the boat is v = at = 3.00 m/s2 * 15.0 s = 45.0 m/s. The frequency of the sound wave heard by the person on the boat is then shifted down by an amount proportional to the relative velocity, or f = 755 Hz - (45.0 m/s / 343 m/s) * 755 Hz = 745.7 Hz.
To solve this problem, we need to consider the Doppler effect, which describes how the frequency of a sound wave changes depending on the relative motion between the source of the sound and the observer.
The formula for the Doppler effect is:
f' = f * (v + vo) / (v + vs)
where,
f' = observed frequency
f = actual frequency emitted by the source
v = speed of sound in air
vo = velocity of the observer relative to the medium
vs = velocity of the source relative to the medium
In this case, the speedboat is moving away from the dock, so the observer (person on the boat) is moving away from the source (siren on the dock).
Given information:
f = 755 Hz (actual frequency emitted by the siren)
vo = 0 m/s (since the person is on the boat)
v = speed of sound in air = 343 m/s (at 20.0°C)
vs = velocity of the speedboat
To find vs, we will use the kinematic equation:
v = u + at
where,
v = final velocity
u = initial velocity
a = acceleration
t = time
In this case, the initial velocity (u) is 0 m/s, as the speedboat starts from rest. We are given the acceleration (a) as +3.00 m/s^2. We need to find the final velocity (v) when the displacement from the dock is +45.0 m.
Using the kinematic equation, we have:
v = u + at
v = 0 + 3.00 * t
Since the boat starts from rest, the equation simplifies to:
v = 3.00 * t
To find the time (t) it takes for the boat to reach the displacement of +45.0 m, we can use the kinematic equation once again:
s = ut + 0.5 * a * t^2
where,
s = displacement
u = initial velocity
a = acceleration
t = time
Plugging in the known values:
45.0 = 0 * t + 0.5 * 3.00 * t^2
Simplifying the equation:
45.0 = 1.5 * t^2
Divide both sides by 1.5:
t^2 = 30.0
Taking the square root of both sides:
t ≈ 5.48 s
Now that we have the time taken (t), we can calculate the final velocity (v):
v = 3.00 * t
v ≈ 3.00 * 5.48
v ≈ 16.44 m/s
Now we have the final velocity (v) of the speedboat, which is the velocity of the source (vs) in the Doppler effect formula.
Using the Doppler effect formula:
f' = f * (v + vo) / (v + vs)
f' = 755 * (343 + 0) / (343 + 16.44)
Simplifying the equation:
f' ≈ 711.82 Hz
Therefore, the frequency of the sound heard by a person on the boat, when the boat's displacement from the dock is +45.0 m, is approximately 711.82 Hz.
To solve this problem, we need to use the Doppler effect equation:
f' = f (v + vb) / (v + vs)
Where:
f' is the observed frequency (heard by the person on the boat)
f is the emitted frequency (produced by the siren on the dock)
v is the speed of sound in air
vb is the velocity of the boat relative to the air
vs is the velocity of sound relative to the air
We know that:
f = 755 Hz (emitted frequency)
v = 343 m/s (speed of sound in air at 20.0°C, you can look this up in a table)
vb = 45.0 m (displacement of the boat from the dock)
vs = 0 m/s (since the velocity of sound relative to the air is zero)
Substituting the given values into the Doppler effect equation:
f' = (755 Hz) × (343 m/s + 45.0 m) / (343 m/s + 0 m/s)
Simplifying:
f' = (755 Hz) × (388 m/s) / (343 m/s)
f' = 853 Hz
Therefore, the frequency of the sound heard by a person on the boat when the boat's displacement from the dock is +45.0 m is approximately 853 Hz.