Question

A motorcycle starts from rest and accelerates along a straight path at $2\,m/{s^2}$. At the straight point of the motorcycle, there is a stationary electric siren. How far has the motorcycle gone when the driver hears the frequency of the siren at $94\%$of its value when the motorcycle was at rest?
A) 49 m
B) 98 m
C) 147 m
D) 196 m

Answer 1

Use the Doppler shift formula to determine the speed at which the driver must be moving to hear the frequency at $94\%$ of its value. For the given amount of acceleration, use the equation of uniform motion to determine the distance travelled when the motorcycle has the speed calculated previously.

Formula used:
-Observed frequency due to Doppler shift: ${f_0} = \dfrac{{v - {v_0}}}{v}{f_s}$ where ${f_0}$ is the frequency of the sound heard by the motorcycle, $v = 330\,m/s$ is the velocity of sound in air, ${v_0}$ is the velocity of the motorcycle.
-${v_0}^2 = 2ad$ where $a$ is the acceleration of the motorcycle and $d$ is the distance it has traveled.

Complete step by step solution:
We’ve been given that the driver hears the frequency of the siren as $94\%$ of its original value. So using the Doppler shift formula, we can write
$\Rightarrow \dfrac{{94}}{{100}} = \dfrac{{330 - {v_0}}}{{330}}$
Multiplying both sides by 330, we get
$\Rightarrow 330 - {v_0} = 310.2$
$\Rightarrow {v_0} = 19.8\,m/s$
So the motorcycle must be moving at this speed to hear the siren having a frequency $94\%$ of its original frequency. Now, let’s use the equation of motion to determine the distance travelled by the motorcycle under constant acceleration to reach this speed. So, using
$\Rightarrow {v_0} = \sqrt {2ad}$
On substituting the value of ${v_0} = 19.8\,m/s$ and $a = 2\,m/{s^2}$, we get
$\Rightarrow {19.8^2} = 4d$
Dividing both sides by 4, we get
$d = 98\,m$ which corresponds to the option (B).

Note:
The problem combines the principles of Doppler shift in frequencies and uniform motion due to constant acceleration and we should the basic formula of these concepts. Since the driver hears a frequency that is lower than the actual frequency, it is moving in a direction away from the speaker and we should be careful in using the correct direction of velocity in the Doppler shift formula.