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?
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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.
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