Question

An ambulance with a siren emitting a whine at 1600Hzovertakes and passes a cyclist pedaling a bike at 2.44m/s. After being passed, the cyclist hears a frequency of 1590Hz. How fast is the ambulance moving?

Short answer

The ambulance is moving at 4.61m/s

Step-by-step solution

The given data

1. The frequency of the siren is, f = 1600Hz.
2. The speed of the cyclist is, v_D = 2.44m/s.
3. Frequency of the siren heard by the cyclist is, f' = 1590Hz.

Understanding the concept of the Doppler Effect

Inserting the given values in the formula obtained from Doppler’s effect, we can find the speed of the ambulance.

Formula:

The frequency that is received by the listener as per the Doppler Effect,

$\mathit{f}\mathbf{\text{'}}\mathbf{=}\mathit{f}\frac{\left(v+v_D\right)}{\left(v+v_s\right)}$ …(i)

Calculation of the speed of ambulance

We know that the speed of sound is v = 343m/s

Using the formula of equation (i) and substitute all the value in this equation, the speed of the ambulance is given as:

$\begin{array}{rcl}1590Hz& =& 1600Hz\times \frac{\left(343m/s+2.44m/s\right)}{\left(343m/s+v_s\right)}\\ v_s& =& 4.6125m/s\\ & =& 4.61m/s\end{array}$

Therefore, the speed of the ambulance is 4.61m/s.