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

The vehicle's speed when the driver hears the siren at 94% of its original frequency is determined from the Doppler effect for a moving observer receding from a stationary source: $f' = f \left( \frac{v_s - v_o}{v_s} \right)$.

Given $f' = 0.94f$ and $v_s = 330\,\text{m/s}$, solve for $v_o$:

$0.94 = \frac{330 - v_o}{330}$
$v_o = 330 - 0.94 \times 330 = 19.8\,\text{m/s}$

With constant acceleration $a = 2\,\text{m/s}^2$ from rest, the distance traveled to reach this velocity is found from $v^2 = u^2 + 2ax$:

$19.8^2 = 0 + 2 \cdot 2 \cdot x$
$392.04 = 4x$
$x = 98.01\,\text{m}$

Rounding to the nearest meter gives $98\,\text{m}$, corresponding to option A.