Question

A motorcycle starts from rest and accelerates along a straight line at $2.2{\text{m/s}}^2.$ The speed of sound is $330\text{m/s}$. A siren at the starting point remains stationary. When the driver hears the frequency of the siren at 90% of the frequency heard when the motorcycle is stationary, the distance travelled by the motorcyclist from his initial position is

• A. $123.75\text{m}$
• B. $247.5\text{m}$
• C. $495\text{m}$
• D. $990\text{m}$

Answer 1

The correct option is B $247.5\text{m}$

Let $v_m$ be the velocity of motorcyclist at that instant and $v$ be velocity of sound.

Given apparent frequency heard by motorcyclist,
$f^{\prime }=\frac{90}{100}{f}_0$
where $f_0$ is the frequency heard when motorcycle is stationary.

Also, let the velocity of motor cycle at that point be $v_m$. Using equation of Doppler effect for moving listener,
$f^{\prime }=f_0\frac{v-v_m}{v}$

$\Rightarrow f^{\prime }=\frac{90}{100}{f}_0=f_0\left(\frac{v-v_m}{v}\right)$
$\Rightarrow 9v=10v-10v_m\Rightarrow v_m=\frac{v}{10}\dots \left(1\right)$

Now, using kinematic equation for motorcyclist,
$v_m^2-u_m^2=2as$
$\Rightarrow v_m^2-0=2as=\frac{v^2}{100}$
[from (1)]

or $s=\frac{v^2}{200a}=\frac{{330}^2}{200\times 2.2}=247.5\text{m}$
is the distance travelled by motorcyclist from initial position.