A stationary source is emmitting sound at a fixed frequency `f_(0)`, which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is `1.2%` of `f_(0)`. What is the difference in the speeds of the cars (in km per hour) to the nearest integer ? The cars are moving at constant speeds much smaller than the speed of sound which is `330 ms^(-1)`.

Explanation

Let `v` be the speed of sound and `v_(c)` and `f_(0)` the speed and frequency of car.
The frequency of sound reflected by the car is
:. `f'_(1) = f_(0) [(v + v_(c))/(v - v_(c))]`
Differenting the above equation w.r.t. `v_(c)`, we get
`(df'_(1))/(dv_(c)) = f_(0)[(2v)/((v - v_(c))^(2))] = (f_(0))(2v)/(v^(2))` (`because` `v_(c) ltlt v`)
:. `((df'_(1))/(f_(0)) xx 100 = (2)/(v) xx dv_(c)`
:. `0.012 = (2 xx dv_(c))/(330)` :. `dv_(c) = 0.198 m//s ~~ 7km//h`