A speed boat at rest begins to move in a straight line from a dock. The boat has a constant acceleration of +3.0 m/s squared. Attached to the dock is a siren that is producing a 755 Hz tone. If the air temperature is 20 degrees C, what is the frequency of the sound heard by a person on the boat when the boat's displacement from the dock is +45.0m?
To determine the frequency of the sound heard by a person on the boat, we need to take into account the concept of the Doppler effect. The Doppler effect describes the change in frequency or pitch that occurs when a sound source and the observer are in relative motion.
In this scenario, the boat is moving away from the dock, so the sound waves emitted by the siren will be stretched out, resulting in a decrease in frequency or a lower pitch. The formula that relates the observed frequency (f') to the actual frequency (f) is:
f' = f[(V + Vr) / (V + Vs)]
Where:
- f' is the observed frequency
- f is the actual frequency of the sound source
- V is the speed of sound in air
- Vr is the velocity of the receiver (person on the boat) relative to the medium (air)
- Vs is the velocity of the source (siren) relative to the medium
First, let's calculate the velocity of the boat when it has reached a displacement of +45.0m. We can use the SUVAT equation for linear motion:
v^2 = u^2 + 2as
Where:
- v is the final velocity (unknown)
- u is the initial velocity (0 m/s, as the boat started from rest)
- a is the acceleration (3.0 m/s^2)
- s is the displacement (+45.0m)
Substituting the values, we have:
v^2 = 0^2 + 2(3.0)(+45.0)
v^2 = 270.0
v ≈ sqrt(270.0)
v ≈ 16.43 m/s (approximately)
Now, let's calculate the velocity of sound in air at 20 degrees C. We can use the equation:
V = 331.4 + 0.6T
Where:
- V is the velocity of sound
- T is the temperature in degrees Celsius
Substituting the value, we have:
V = 331.4 + 0.6(20)
V ≈ 343.4 m/s (approximately)
Next, we need to calculate the relative velocity of the receiver (person on the boat) and the source (siren). In this case, since the boat is moving away from the dock, the relative velocity is simply the velocity of the boat:
Vr = 16.43 m/s
Finally, substitute the values into the Doppler effect equation:
f' = f[(V + Vr) / (V + Vs)]
f' = 755[(343.4 + 16.43) / (343.4 + 0)]
f' = 755(359.83 / 343.4)
f' ≈ 789.8 Hz (approximately)
Therefore, the frequency of the sound heard by a person on the boat when the boat's displacement is +45.0m is approximately 789.8 Hz.