Two small planes (we label them A and B) are traveling along the same axis (X ax

Question

Two small planes (we label them A and B) are traveling along the same axis (X axis). At the time t=0s, plane A is at the coordinate x=0 and plane B is at the coordinate x=50km. Both planes started moving at t=0s. Plane A is traveling at a velocity Va = (450km/h)x (x is a vector) and B is traveling at a velocity Vb = -(350km/h)x (x is a vector).
a) At the time t=0s plane A emits a siren (a sound wave) at frequency f=3kHz. We assume that the sound of the siren reaches B. What is the frequency of the sound that a person in B will hear?
b) If instead of A, at time t=0s, plane B is the one emitting the siren at f=3kHz. We assume that the sound of the siren reaches A. What is the frequency of the sound that a person in A will hear?
c) If we wait until t=5min then plane A (and not B) emits the siren at frequency f=3kHz. We assume that the sound of the siren reaches B. What is the frequency of the sound that a person in B will hear?
d) If we wait until t=5min then plane B (and not A) emits the siren at f=3kHz. We assume that the sound of the siren reaches A. What is the frequency of the sound a person in A will hear?

Explanation / Answer

Va = 450 * 5/18 = 125 m/s (x directn)
Vb = -(350)*5/18 = 97.22 m/s (x direction )

Assuming speed of sound = 344 m/s
1> f = 3*(344 + 97.22)/(344-125) = 6.044 khz
2> f = 3 * (344 + 125)/(344-97.22) = 5.70 Khz
After 5 mins , checking the directions of motion of planes :
taking plane B as reference vel of plane a w.r.t b = 125 + 97.22 = 222.22 m/s

after 5 mins : distance= 222.22(5*60)m= 66.666km which is greater than 5 km i.e. planes are now moving in opposite directions.

3> f = 3 * (344-97.22)/(344+125) = 1.578 khz

4> f = 3*(344-125)/(344+97.22) = 1.489 khz

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