Question
A car starts from rest, accelerating at 6.0 m/s for 5s. It continues at this speed for 25s, and then comes to a stop in 4s. Calculate (a) total distance covered, and (b) the average velocity for the whole displacement.
Answers
GPT 3.5
To solve this problem, we can break it down into three parts: the initial acceleration, the constant velocity, and the deceleration.
(a) Total distance covered:
For the initial acceleration, we can use the formula:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the car starts from rest, the initial velocity is 0 m/s. Therefore, the distance covered during the initial acceleration is:
distance = (0 * 5) + (0.5 * 6.0 * 5^2)
distance = 0 + (0.5 * 6.0 * 25)
distance = 0 + 75
distance = 75 meters
During the constant velocity phase, the car is traveling at 6.0 m/s for 25 seconds, so the distance covered is:
distance = velocity * time
distance = 6.0 * 25
distance = 150 meters
Finally, during the deceleration phase, the car comes to a stop in 4 seconds. We can use the same formula as before, but with a negative acceleration value since the car is decelerating:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
distance = (6.0 * 4) + (0.5 * -6.0 * 4^2)
distance = 24 + (-24)
distance = 0 meters
The total distance covered is the sum of the distances covered during each phase:
Total distance = 75 + 150 + 0
Total distance = 225 meters
(b) Average velocity for the whole displacement:
The average velocity is given by the formula:
average velocity = total displacement / total time
The total displacement is the change in position from the initial point to the final point, which is 0 meters since the car starts and ends at the same position.
The total time is the sum of the times for each phase:
Total time = initial acceleration time + constant velocity time + deceleration time
Total time = 5 + 25 + 4
Total time = 34 seconds
Therefore, the average velocity for the whole displacement is:
average velocity = total displacement / total time
average velocity = 0 / 34
average velocity = 0 m/s
(a) Total distance covered:
For the initial acceleration, we can use the formula:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the car starts from rest, the initial velocity is 0 m/s. Therefore, the distance covered during the initial acceleration is:
distance = (0 * 5) + (0.5 * 6.0 * 5^2)
distance = 0 + (0.5 * 6.0 * 25)
distance = 0 + 75
distance = 75 meters
During the constant velocity phase, the car is traveling at 6.0 m/s for 25 seconds, so the distance covered is:
distance = velocity * time
distance = 6.0 * 25
distance = 150 meters
Finally, during the deceleration phase, the car comes to a stop in 4 seconds. We can use the same formula as before, but with a negative acceleration value since the car is decelerating:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
distance = (6.0 * 4) + (0.5 * -6.0 * 4^2)
distance = 24 + (-24)
distance = 0 meters
The total distance covered is the sum of the distances covered during each phase:
Total distance = 75 + 150 + 0
Total distance = 225 meters
(b) Average velocity for the whole displacement:
The average velocity is given by the formula:
average velocity = total displacement / total time
The total displacement is the change in position from the initial point to the final point, which is 0 meters since the car starts and ends at the same position.
The total time is the sum of the times for each phase:
Total time = initial acceleration time + constant velocity time + deceleration time
Total time = 5 + 25 + 4
Total time = 34 seconds
Therefore, the average velocity for the whole displacement is:
average velocity = total displacement / total time
average velocity = 0 / 34
average velocity = 0 m/s