A car accelerates from rest at 2.0 m/s2 for 20 s. The speed then remains constant for 20 s, after which the car slows down with an acceleration of 3 m/s2 until the car stops. What is the total distance traveled?
first part
final v = a t = 40 m/s
average v = 20 m/s
x = 0 + 20(20) = 400 m
second part
v = 40
x = 400 + 40(20) = 1200 m
third part
Vi = 40
average v = 20
time = 40/3 = 13.3
x = 1200+20(13.3) = 1466 m
3066
To find the total distance traveled by the car, we need to break down the motion of the car into different segments and calculate the distances for each segment individually. Let's go step by step.
Step 1: Calculate the distance traveled during the first segment of the motion, where the car is accelerating.
We know that the car starts from rest, so the initial velocity (u) is 0 m/s. The car accelerates at a rate of 2.0 m/s^2 for 20 seconds. We can use the following kinematic equation to calculate the distance (s1):
s1 = u * t + (1/2) * a * t^2
Plugging in the values, we have:
s1 = 0 * 20 + (1/2) * 2.0 * (20^2)
s1 = 0 + 0.5 * 2.0 * 400
s1 = 0 + 400
s1 = 400 meters
Step 2: Calculate the distance traveled during the second segment, where the car maintains a constant speed.
During this segment, the car travels at a constant speed for 20 seconds. Since the speed remains constant, the distance traveled (s2) can be calculated using the formula:
s2 = v * t
Given that the speed (v) is constant, we can use any value within that time period. Let's choose the final velocity after acceleration, which is:
v = u + a * t
v = 0 + 2.0 * 20
v = 0 + 40
v = 40 m/s
Now calculate s2:
s2 = 40 * 20
s2 = 800 meters
Step 3: Calculate the distance traveled during the third segment, where the car decelerates.
In this segment, the car decelerates with an acceleration of 3.0 m/s^2 until it stops. Let's calculate the time it takes for the car to come to a stop using the formula:
v = u + a * t
0 = v + (-3.0) * t
Rearranging the equation, we have:
3.0t = v
t = v / 3.0
Plugging in the value of the speed after the constant speed segment (40 m/s):
t = 40 / 3.0
t ≈ 13.33 seconds
Now we can calculate the distance traveled during this segment (s3) using the formula:
s3 = u * t + (1/2) * a * t^2
Plugging in the values, we have:
s3 = 40 * 13.33 + (1/2) * (-3.0) * (13.33^2)
s3 ≈ 533.2 - 532.75
s3 ≈ 0.45 meters
Step 4: Calculate the total distance traveled.
To calculate the total distance, we simply add up the distances from each segment:
Total distance = s1 + s2 + s3
Total distance = 400 + 800 + 0.45
Total distance ≈ 1200.45 meters
Therefore, the total distance traveled by the car is approximately 1200.45 meters.