An electric vehicle starts from rest and accelerates at a rate of 3.0 m/s2 in a straight line until it reaches a speed of 26 m/s. The vehicle then slows at a constant rate of 1.0 m/s2 until it stops.
a)How much time elapses from start to stop?
b)How far does the vehicle travel from start to stop?
a) Add up the times it takes to accelerate and decelerate.
The time required to accelerate is
(26 m/s)/3 m/s^2 = 8.67 s
The time required to decelerate from 26 m/s^2 at a constant rate of -1 m/s^2 is 26 s.
b) Add up the distances covered while accelerating and decelerating. Each equals the average velocity (13 m/s) times the time interval.
Thank you.
To find the time elapsed from start to stop, we need to calculate the time it takes for the vehicle to accelerate to its final speed and the time it takes for the vehicle to decelerate and come to a stop.
a) To find the time it takes for the vehicle to accelerate, we can use the formula:
final velocity (v) = initial velocity (u) + acceleration (a) × time (t)
In this case, the initial velocity is 0 m/s (since the vehicle starts from rest), the final velocity is 26 m/s, and the acceleration is 3.0 m/s². Let's solve for time (t):
26 m/s = 0 m/s + 3.0 m/s² × t
Rearranging the equation, we get:
t = (26 m/s - 0 m/s) / 3.0 m/s²
t = 26 m/s / 3.0 m/s²
t ≈ 8.67 s
So, the time it takes for the vehicle to accelerate to its final speed is approximately 8.67 seconds.
To find the time it takes for the vehicle to decelerate and come to a stop, we can use the same formula, but with the deceleration as the negative acceleration:
0 m/s = 26 m/s - 1.0 m/s² × t
Rearranging the equation, we get:
t = (26 m/s - 0 m/s) / 1.0 m/s²
t = 26 m/s / 1.0 m/s²
t = 26 s
So, the time it takes for the vehicle to decelerate and come to a stop is 26 seconds.
Now, we can calculate the total elapsed time by adding the time it takes for acceleration and the time it takes for deceleration:
Total elapsed time = Time for acceleration + Time for deceleration
Total elapsed time = 8.67 s + 26 s
Total elapsed time ≈ 34.67 s
Therefore, the total elapsed time from start to stop is approximately 34.67 seconds.
b) To find the distance traveled by the vehicle from start to stop, we need to calculate the distance covered during acceleration and the distance covered during deceleration.
The distance covered during acceleration can be calculated using the equation:
distance (d) = initial velocity (u) × time (t) + (1/2) × acceleration (a) × time (t)².
In this case, the initial velocity is 0 m/s, the time for acceleration is 8.67 s, and the acceleration is 3.0 m/s². Let's solve for distance (d):
distance = 0 m/s × 8.67 s + (1/2) × 3.0 m/s² × (8.67 s)²
distance = 0 + (1/2) × 3.0 m/s² × 75.3369 s²
distance ≈ 113.0045 m
So, the distance covered during acceleration is approximately 113 meters.
The distance covered during deceleration can be calculated using the same formula, but with the deceleration as the negative acceleration:
distance = final velocity (v) × time (t) + (1/2) × acceleration (a) × time (t)²
In this case, the final velocity is 0 m/s, the time for deceleration is 26 s, and the acceleration is -1.0 m/s². Let's solve for distance (d):
distance = 0 m/s × 26 s + (1/2) × (-1.0 m/s²) × (26 s)²
distance = 0 + (1/2) × (-1.0 m/s²) × 676 s²
distance ≈ -17524 m
The negative sign indicates that the distance is in the opposite direction. However, in this case, we can consider the magnitude of distance (as the vehicle is moving in a straight line) and ignore the negative sign.
So, the distance covered during deceleration is approximately 17524 meters.
Now, we can calculate the total distance:
Total distance = Distance during acceleration + Distance during deceleration
Total distance = 113 m + 17524 m
Total distance ≈ 17637 m
Therefore, the vehicle travels approximately 17637 meters from start to stop.