A motor cycle starts from rest and accelerates along a straight path at $$2\,m/{s^2}.$$  At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at $$94\% $$  of its value when the motor cycle was at rest? (Speed of sound = $$330\,m{s^{ - 1}}$$  )

Solution :

$$\eqalign{ & v_m^2 - {u^2} = 2\,as \cr & \Rightarrow \,\,v_m^2 = 2 \times 2 \times s \cr & \therefore \,\,{v_m} = 2\sqrt s \cr} $$
According to Doppler’s effect
$$\eqalign{ & 0.94\,v = v\left[ {\frac{{330 - 2\sqrt s }}{{330}}} \right] \cr & \Rightarrow \,\,s = 98.01\,m \cr} $$