An electric vehicle starts from rest and accelerates at a rate of 2.3 m/s2 in a straight line until it reaches a speed of 17 m/s. The vehicle then slows at a constant rate of 1.4 m/s2 until it stops. (a) How much time elapses from start to stop? (b) How far does the vehicle move from start to stop?
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To find the answer to these questions, we can use the equations of motion.
(a) To determine the time elapsed from start to stop, we need to consider the two phases of motion separately: the acceleration phase and the deceleration phase.
The equation to calculate the time taken during acceleration is:
v = u + at
where:
v = final velocity
u = initial velocity (zero in this case since the vehicle starts from rest)
a = acceleration
Given:
v = 17 m/s
a = 2.3 m/s^2
Plugging in these values into the equation, we can solve for the time taken during acceleration:
17 = 0 + 2.3t
Simplifying the equation gives:
2.3t = 17
t = 17 / 2.3
t ≈ 7.39 seconds
So, the time taken during the acceleration phase is approximately 7.39 seconds.
Now, let's calculate the time taken during the deceleration phase.
Using the same equation as before, but this time with opposite acceleration (since it's slowing down), we have:
0 = 17 + (-1.4)t
Simplifying and solving for t, we get:
-1.4t = -17
t ≈ 17 / 1.4
t ≈ 12.14 seconds
Thus, the time taken during the deceleration phase is approximately 12.14 seconds.
To find the total time elapsed from start to stop, we add the time taken during the acceleration and deceleration phases:
Total time = Time during acceleration + Time during deceleration
Total time ≈ 7.39 + 12.14 ≈ 19.53 seconds
Therefore, the total time elapsed from start to stop is approximately 19.53 seconds.
(b) To determine the distance the vehicle moved from start to stop, we can use another equation of motion:
s = ut + 0.5at^2
where:
s = distance
u = initial velocity (zero in this case since the vehicle starts from rest)
a = acceleration
t = time
Since the initial velocity is zero and the vehicle comes to a stop after both phases, the total distance can be calculated as the sum of the distances covered during acceleration and deceleration:
Distance = Distance during acceleration + Distance during deceleration
For the acceleration phase:
s1 = 0.5at1^2
s1 = 0.5 * 2.3 * (7.39)^2
For the deceleration phase:
s2 = 0.5at2^2
s2 = 0.5 * (-1.4) * (12.14)^2
Total distance = s1 + s2
Total distance ≈ 79.13 + (-106.62) ≈ -27.49 meters
Note: The negative sign indicates that the vehicle moved in the opposite direction of the initial motion.
Therefore, from start to stop, the vehicle moves approximately 27.49 meters in the opposite direction.