A source of sound moves toward you at speed V and away from Jane, who is standing on the other side of it.
You hear the sound at twice the frequency as Jane. What is the speed of the source? Assume that the speed of sound is 340 m/s.
Note that the source of the sound is a moving source. With this, the relationship between the speed of source of the sound and the speed of the sound itself can be described by the formula of Doppler Effect.
`f ' = v (f/(v+-u_s))`
where f ' - the apparent frequency (heard by the observer)
f - frequency of source
`u_s`- speed of source
v - speed of sound
Let `f_j ` the apparent frequency heard by Jane.
`f_j = v(f/(v+u_s))`
`u_s` is positive since the source is moving away from Jane. Then, susbtitute v=340 m/s.
`f_j = 340(f/(340+u_s))`
`f_j = (340f)/(340+u_s)`
This is the equation of the apparent frequency heard by Jane.
Then, let `f_I` the apparent frequency that I hear.
`f_I = v(f/(v-u_s))`
`u_s ` is negative since the source is moving toward me. Again, subsitute v=340 m/s.
`f_I = 340(f/(340-u_s))`
Since the frequency that I hear is twice of Jane's, hence `f_I = 2f_j` .
`2 f_j = (340f)/(340-u_s)`
Substiute `f_j = (340f)/(340+u_s)`
`2[(340f)/(340+u_s)] = (340f)/(340-u_s)`
To simplify, divide both sides by 340f.
`2/(340+u_s) = 1/(340-u_s)`
`2(340-u_s) = 340+u_s`
`680- 2u_s = 340 + u_s`
`680- 340 = u_s+2u_s`
`340 = 3u_s`
`113.3 = u_s`
Hence, the speed of the source is 113.3 m/s.