Question

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 \(=330 \mathrm{~m} / \mathrm{s}\) )

• A. 98 m
• B. 49 m
• C. 196 m
• D. 24.5 m

Hint:

The vehicle moves away from the siren, so use the Doppler formula for a moving observer and then the equation of motion from rest.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The vehicle moves away from the stationary siren, so the observer hears a lower frequency: \(f'=f\left(\dfrac{v-v_o}{v}\right)\).
Its speed at that moment follows from \(v_o^2=2as\).

Step 2: Find the Speed of the Vehicle
\(f'=0.94f\), so \(\dfrac{v-v_o}{v}=0.94\).
\(v_o=0.06\times 330=19.8\text{ m/s}\)

Step 3: Find the Distance
\(s=\dfrac{v_o^2}{2a}=\dfrac{(19.8)^2}{2\times 2}=\dfrac{392.04}{4}\approx 98\text{ m}\)

Step 4: Compare the options
The distance of about 98 m matches the first option.

Final Answer:
The vehicle has gone about 98 m. \[ \boxed{\text{(A) }98\text{ m}} \]