A bat locates insects by emitting ultrasonic 'chirps' and then listening for echoes from the bugs. Suppose a bat chirp has a frequency of 25 kHz. How fast would the bat have to fly, and in what direction, for you to just barely be able to hear the chirp at 20 kHz?

Verified step by step guidance

1. Identify the phenomenon involved: This problem involves the Doppler Effect, which describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave.
2. Determine the original frequency (_{ ext{original}}) and the observed frequency (_{ ext{observed}}): In this case, _{ ext{original}} = 25 kHz and _{ ext{observed}} = 20 kHz.
3. Use the Doppler Effect formula for sound when the source is moving towards a stationary observer: _{ ext{observed}} = _{ ext{original}} imes rac{v + v_{ ext{source}}}{v}, where _{ ext{observed}} is the frequency heard by the observer, _{ ext{original}} is the emitted frequency, v is the speed of sound in air (approximately 340 m/s), and v_{ ext{source}} is the speed of the bat.
4. Solve for the bat's speed (_{ ext{source}}): Rearrange the formula to isolate _{ ext{source}}. This will involve algebraic manipulation to solve for _{ ext{source}} in terms of _{ ext{observed}}, _{ ext{original}}, and v.
5. Determine the direction of the bat's flight: Since the observed frequency is lower than the emitted frequency, the bat must be flying away from the observer.

Key Concepts

Doppler Effect

The Doppler Effect is a phenomenon observed when there is a relative motion between a source of sound and an observer. When the source moves towards the observer, the frequency of the sound waves increases, resulting in a higher pitch. Conversely, if the source moves away, the frequency decreases, leading to a lower pitch. This effect is crucial for understanding how the bat's movement affects the frequency of the chirp heard by an observer.

Frequency and Pitch

Frequency refers to the number of cycles of a wave that occur in a unit of time, typically measured in hertz (Hz). In sound, frequency is directly related to pitch; higher frequencies correspond to higher pitches. In this scenario, the bat emits a chirp at 25 kHz, and the observer hears it at 20 kHz, indicating a change in frequency due to the bat's motion, which is essential for solving the problem.

Speed of Sound

The speed of sound is the rate at which sound waves propagate through a medium, typically air, and is approximately 343 meters per second at room temperature. This speed is a critical factor in calculating how fast the bat must fly to alter the frequency of its chirp to the level that can be heard by the observer. Understanding the speed of sound allows us to apply the Doppler Effect equations effectively in this context.