To determine how long the observer will hear the horn after the car stops sounding it, we need to break down the problem into manageable steps. We'll analyze the car's motion, calculate its velocity at the moment the horn stops, and then figure out how long it takes for the sound to reach the observer.
Step 1: Calculate the Car's Velocity After 20 Seconds
The car starts from rest and accelerates at a rate of 1.1 m/s². We can use the formula for velocity under constant acceleration:
v = u + at
Where:
- v = final velocity
- u = initial velocity (0 m/s, since the car starts from rest)
- a = acceleration (1.1 m/s²)
- t = time (20 s)
Plugging in the values:
v = 0 + (1.1 m/s²)(20 s) = 22 m/s
Step 2: Determine the Distance Covered by the Car in 20 Seconds
Next, we need to find out how far the car travels in those 20 seconds. We can use the formula for distance under constant acceleration:
s = ut + (1/2)at²
Substituting the known values:
s = (0)(20) + (1/2)(1.1)(20²)
s = 0 + (0.55)(400) = 220 m
Step 3: Calculate the Remaining Distance to the Observer
The initial distance from the car to the observer was 330 m. After 20 seconds, the car has traveled 220 m, so the remaining distance is:
Remaining distance = 330 m - 220 m = 110 m
Step 4: Time for the Sound to Reach the Observer
Once the car stops sounding the horn, we need to calculate how long it takes for the sound to travel the remaining distance of 110 m. The speed of sound in air is given as 330 m/s. We can use the formula:
time = distance / speed
Substituting the values:
time = 110 m / 330 m/s = 1/3 s
Step 5: Total Duration the Observer Hears the Horn
The observer hears the horn for the entire 20 seconds while the car is sounding it, plus the additional time it takes for the sound to reach them after the horn stops:
Total time = 20 s + 1/3 s = 20 1/3 s
To express this in a more manageable form, we can convert 1/3 seconds into a fraction:
Total time = 20 + 0.333... = 20.333... seconds
Final Result
The observer will hear the horn for a total of 62/3 seconds, which is approximately 20.33 seconds. This includes the time the horn was actively sounding and the time it took for the sound to travel the remaining distance to the observer.
