A car is initially at rest, 330 m away from a stationary observer. It begins to move towards the observer with an acceleration of `1.1 m s^(-2)` sounding its horn continuously. 20 s later, the driver stops sounding the horn. The velocity of sound in air is `330 m s^(-1)`. The observer will hear the sound of the horn for a duration of

Options

A. 20 s
B. 21 s
C. `20 2/3 s`
D. `19 1/3 s`

Answer

To solve the problem step by step, we need to determine how long the observer will hear the sound of the horn after the car starts moving towards them.

### Step 1: Calculate the displacement of the car in 20 seconds
The car starts from rest and accelerates towards the observer. The formula for displacement under constant acceleration is given by:

\[ s = ut + \frac{1}{2} a t^2 \]

Where:
- \( s \) = displacement
- \( u \) = initial velocity (0 m/s, since the car starts from rest)
- \( a \) = acceleration (1.1 m/s²)
- \( t \) = time (20 s)

Substituting the values:

\[ s = 0 \cdot 20 + \frac{1}{2} \cdot 1.1 \cdot (20)^2 \]
\[ s = \frac{1}{2} \cdot 1.1 \cdot 400 \]
\[ s = 220 \text{ m} \]

### Step 2: Determine the new distance of the car from the observer after 20 seconds
Initially, the car was 330 m away from the observer. After 20 seconds, the car has moved 220 m closer.

\[ \text{New distance} = 330 \text{ m} - 220 \text{ m} = 110 \text{ m} \]

### Step 3: Calculate the time taken for the sound to reach the observer
The sound travels at a speed of 330 m/s. The time taken for the sound to travel 110 m is given by:

\[ t = \frac{\text{distance}}{\text{speed}} = \frac{110 \text{ m}}{330 \text{ m/s}} = \frac{1}{3} \text{ s} \]

### Step 4: Calculate the total time the observer hears the horn
The horn is sounded for 20 seconds, and the sound takes an additional \( \frac{1}{3} \) seconds to reach the observer after the horn stops.

Thus, the total time \( T \) that the observer hears the horn is:

\[ T = 20 \text{ s} + \frac{1}{3} \text{ s} = 20 \frac{1}{3} \text{ s} \]

### Final Answer
The observer will hear the sound of the horn for a duration of \( 20 \frac{1}{3} \) seconds.

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