Answer:
229 m/s
Explanation:
The Doppler effect is the phenomenon in which the sound frequency heard by an observer changes depending on the relative velocity of the source. The frequency of a sound wave is higher when the source is moving towards the receiver, and lower when the source is moving away. This is described by the Doppler equation:
\(\huge \text {$ f_r=f_s\ $}\Huge \text {$ \frac{c+v_r}{c+v_s} $}\)
where:
- \(\large \text {$ f_r $}\) is the frequency heard by the receiver
- \(\large \text {$ f_s $}\) is the frequency emitted at the source
- \(\large \text {$ c $}\) is the speed of sound
- \(\large \text {$ v_r $}\) is the velocity of the receiver
- \(\large \text {$ v_s $}\) is the velocity of the source
Note: \(\large \text {$ v_r $}\) is positive if the receiver is moving towards the source, negative if away.
Conversely, \(\large \text {$ v_s $}\) is positive if the receiver is moving away from the source, and negative if towards.
In this case, we are told that the frequency heard by the receiver is one-fifth the frequency emitted at the source.
\(\large \text {$ f_r = f_s / 5 $}\)
We are also told that the planes are moving away from each other at equal speed.
\(\large \text {$ v_r = -v $}\\\large \text {$ v_s=v $}\)
Finally, we are told that the speed of sound is 343 m/s.
\(\large \text {$ c=343\ m/s $}\)
Substituting:
\(\large \text {$ f_s/5=f_s $}\LARGE \text {$ \frac{343-v}{343+v} $}\\\large \text {$ 1/5= $}\LARGE \text {$ \frac{343-v}{343+v} $}\\\large \text {$ 343+v=5(343-v) $}\\\large \text {$ 343+v=1715-5v $}\\\large \text {$ 6v=1372 $}\\\large \text {$ v\approx 229$}\)
The speed of each plane is approximately 229 m/s.
