A bat at rest sends out ultrasonic sound waves at 51.1 and receives them returned from an object moving directly away from it at 22.3 .
To solve this problem, you can use the formula for the Doppler effect, which relates the frequency observed by an observer to the frequency emitted by a source. The formula for the Doppler effect for sound waves is as follows:
f_observed = f_emitted * (v_sound + v_observer) / (v_sound + v_source)
Where:
f_observed = observed frequency
f_emitted = emitted frequency
v_sound = speed of sound in air
v_observer = velocity of the observer
v_source = velocity of the source
In this case, the bat is emitting and receiving ultrasonic sound waves, so the emitted frequency and observed frequency will be the same. We can assume the velocity of the sound in air is 343 meters per second (which is the speed of sound at 20 degrees Celsius).
Using the given values, we have:
f_emitted = f_observed = 51.1 kHz (kilohertz)
v_sound = 343 m/s
v_observer = 0 m/s (since the bat is at rest)
v_source = -22.3 m/s (since the object is moving directly away from the bat)
Substituting the values into the formula, we can solve for the emitted frequency:
51.1 kHz = f_emitted * (343 m/s + 0 m/s) / (343 m/s - 22.3 m/s)
Simplifying the equation, we have:
51.1 kHz = f_emitted * 343 m/s / 320.7 m/s
Dividing both sides of the equation by 343 m/s / 320.7 m/s:
f_emitted = 51.1 kHz / (343 m/s / 320.7 m/s)
Solving this equation, we find:
f_emitted ≈ 54.31 kHz
Therefore, the bat is emitting ultrasonic sound waves at approximately 54.31 kHz.
To calculate the speed of an object moving directly away from a bat, we can use the formula:
Speed = (Speed of sound * Time taken for sound wave to return) / 2
Given:
Speed of sound = 343 m/s (speed of sound in air at room temperature)
Time taken for sound wave to return = 22.3 seconds
Plugging in the values into the formula:
Speed = (343 * 22.3) / 2
Calculating the speed:
Speed = 7618.9 / 2
Speed = 3809.45 m/s
Therefore, the object is moving away from the bat at a speed of approximately 3809.45 m/s.