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Taking into account the Doppler efect, the speed of the bat is closest to 5.02 .
The Doppler effect is defined as the apparent frequency change of a wave produced by the relative movement of the source with respect to its observer. In other words, this effect is the change in the perceived frequency of any wave motion when the sender and receiver, or observer, move relative to each other.
The following expression is considered the general case of the Doppler effect:
Where:
- f1, f: Frequency perceived by the receiver and frequency emitted by the issuer respectively. Its unit of measurement in the International System (S.I.) is the hertz (Hz), which is the inverse unit of the second (1 Hz = 1 s⁻¹)
- v: Velocity of propagation of the wave in the medium. It is constant and depends on the characteristics of the medium. In this case, the speed of sound in air is considered to be 340 m/s
- vR, vE: Speed of the receiver and the sender respectively. Its unit of measure in the S.I. is the m/s
- ±, ∓:
- We will use the + sign:
In the numerator if the receiver approaches the sender
In the denominator if the sender moves away from the receiver
- We will use the sign -:
In the numerator if the receiver moves away from the emitter
In the denominator if the sender approaches the receiver
In this case, a bat emits a sound at a frequency of 3.00×10⁴ Hz= 30,000 Hz as it approaches a wall.
The bat detects beats such that the frequency of the echo is 900 hz higher than the frequency the bat is emitting.
So, you know:
- f1= 30,000 Hz + 900 Hz= 30,900 Hz
- f= 30,000 Hz
- v= 340 m/s
- vE=vR= vbat
- Receiver approaches the sender
Replacing in the expression is considered the general case of the Doppler effect:
Solving:
1.03×( 340 -vbat)= 340
+ vbat
1.03×340 - 1.03×vbat= 340
+ vbat
350.2 - 1.03×vbat= 340
+ vbat
350.2 - 340
= vbat + 1.03×vbat
10.2 = 2.03×vbat
10.2 ÷ 2.03 = vbat
<u><em>5.02 </em></u><u><em>= vbat</em></u>
In summary, the speed of the bat is closest to 5.02 .
Learn more about the Doppler effect: