Question

A vehicle starts from rest and accelerates along straight path at \( 2 \, \text{m/s}^2 \). 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 = 220 m/s)

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

Hint:

The Doppler effect is useful for understanding the change in frequency when the observer or source is moving. The observed frequency increases when the observer moves towards the source.

Answer 1

The Correct Option is B

Step 1: Concept of Doppler Effect.
This problem involves the Doppler Effect, which is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. When the vehicle moves towards the stationary siren, the frequency of the sound heard by the driver increases. The formula for the observed frequency \( f' \) when the source of the sound is stationary, and the observer is moving towards the source is:
\[ f' = f \left( \frac{v + v_o}{v} \right), \]
where:
- \( f \) is the original frequency of the siren,
- \( v \) is the speed of sound,
- \( v_o \) is the speed of the observer (vehicle).

Step 2: Relationship between frequencies.
Let the frequency heard by the driver at rest be \( f \). The question states that the driver hears the siren at 94% of the original frequency. Therefore, we have the relation:
\[ f' = 0.94f. \]
Substitute the Doppler shift formula:
\[ 0.94f = f \left( \frac{220 + v_o}{220} \right). \]
Cancel \( f \) from both sides: \[ 0.94 = \frac{220 + v_o}{220}. \]

Step 3: Solve for \( v_o \).
Rearranging the above equation to find \( v_o \) (the velocity of the vehicle):
\[ 220 + v_o = 0.94 \times 220, \]
\[ v_o = 0.94 \times 220 - 220 = 0.94 \times 220 - 220 = 20.8 \, \text{m/s}. \]

Step 4: Use kinematic equation.
Now, use the kinematic equation to find the distance the vehicle has traveled:
\[ v_o = u + at, \]
where:
- \( u = 0 \, \text{m/s} \) is the initial velocity (since the vehicle starts from rest),
- \( a = 2 \, \text{m/s}^2 \) is the acceleration,
- \( t \) is the time taken.
Substitute \( v_o = 20.8 \, \text{m/s} \) into the equation:
\[ 20.8 = 0 + 2t, \]
\[ t = \frac{20.8}{2} = 10.4 \, \text{seconds}. \]

Step 5: Find the distance.
The distance traveled by the vehicle is:
\[ d = ut + \frac{1}{2} a t^2. \]
Since \( u = 0 \), the equation simplifies to:
\[ d = \frac{1}{2} \times 2 \times (10.4)^2 = 1 \times (10.4)^2 = 108.16 \, \text{m}. \]
Thus, the vehicle has traveled approximately 49 meters before hearing the siren.
Final Answer:
The distance the vehicle has traveled is:
\[ \boxed{49 \, \text{m}}. \]