An automobile accelerates from rest at
2.4 m/s2 for 29 s. The speed is then held
constant for 11 s, after which there is an ac-
celeration of −2.2 m/s2 until the automobile
stops.
What total distance was traveled?
Answer in units of km
To find the total distance traveled by the automobile, we can break down the given information into two parts:
1. The distance traveled during acceleration from rest to constant speed.
2. The distance traveled during deceleration from constant speed to rest.
Part 1: Acceleration from rest to constant speed
During this part, the automobile accelerates at a rate of 2.4 m/s^2 for a duration of 29 seconds.
We can use the equation of motion to find the distance traveled during this time:
distance = (initial velocity)(time) + (0.5)(acceleration)(time^2)
Since the initial velocity is 0 m/s (starting from rest), the equation simplifies to:
distance = (0.5)(acceleration)(time^2)
Plugging in the values, we get:
distance = (0.5)(2.4 m/s^2)(29 s)^2
Calculating this, we find:
distance = 20.828 meters
Part 2: Deceleration from constant speed to rest
During this part, the speed is constant for 11 seconds, followed by deceleration at a rate of -2.2 m/s^2 until the automobile stops.
Since the speed is constant for 11 seconds, the distance traveled during this time is:
distance = speed × time = constant speed × time
To calculate the constant speed, we need to determine the speed reached during the acceleration phase. We can calculate this by multiplying the acceleration by the time it took to reach the constant speed:
speed = acceleration × time = (2.4 m/s^2) × (29 s)
Calculating this, we find:
speed = 69.6 m/s
Now we can calculate the distance traveled during the 11-second duration of constant speed:
distance = speed × time = (69.6 m/s) × (11 s)
Calculating this, we get:
distance = 765.6 meters
Finally, during the deceleration phase until the automobile stops, the distance traveled can also be calculated using the equation of motion:
distance = (initial velocity)(time) + (0.5)(acceleration)(time^2)
Since the initial velocity is the speed obtained during the constant speed phase (69.6 m/s) and the acceleration is -2.2 m/s^2, the equation becomes:
distance = (69.6 m/s)(time) + (0.5)(-2.2 m/s^2)(time^2)
We need to find the time it takes for the automobile to stop. To do this, we can set the equation equal to 0 and solve for time:
(69.6 m/s)(time) + (0.5)(-2.2 m/s^2)(time^2) = 0
Simplifying this equation, we get:
-1.1(time^2) + 69.6(time) = 0
Factoring out time, we get:
time(-1.1(time) + 69.6) = 0
This equation is satisfied when either time = 0 (which is the initial condition) or when -1.1(time) + 69.6 = 0. Solving for time, we find:
-1.1(time) + 69.6 = 0
-1.1(time) = -69.6
time = 69.6 m/s ÷ 1.1 m/s^2
time ≈ 63.273 seconds
Now we can calculate the distance traveled during this time:
distance = (69.6 m/s)(63.273 s) + (0.5)(-2.2 m/s^2)((63.273 s)^2)
Calculating this, we find:
distance ≈ 2204.694 meters
Therefore, the total distance traveled by the automobile is the sum of the distances calculated in Part 1 and Part 2:
total distance = 20.828 meters + 765.6 meters + 2204.694 meters
Converting this to kilometers:
total distance ≈ 2.991122 kilometers
Therefore, the total distance traveled by the automobile is approximately 2.991 km.
Note: The final answer has been rounded to 3 decimal places.