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 24 m/s. The vehicle then slows at a constant rate of 1.5 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?
To answer these questions, we can use the formulas of linear motion. Let's break down the problem step by step.
(a) To find the time elapsed from start to stop, we need to determine the time taken during acceleration and deceleration separately.
For the acceleration phase:
We know the initial velocity (0 m/s), final velocity (24 m/s), and acceleration (2.3 m/s^2). The equation to calculate time during acceleration is:
v = u + at,
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.
Plugging in the values, we have:
24 = 0 + 2.3t.
Solving this equation, we get:
t = 24 / 2.3,
t ≈ 10.43 seconds.
For the deceleration phase:
We know the final velocity (0 m/s), initial velocity (24 m/s in the opposite direction), and acceleration (-1.5 m/s^2). Again, using the same equation:
v = u + at,
0 = 24 + (-1.5)t.
Solving for t:
t = -24 / (-1.5),
t ≈ 16 seconds.
The total time elapsed would be the sum of the acceleration and deceleration times:
Total time = 10.43 + 16,
Total time ≈ 26.43 seconds.
So, the time elapsed from start to stop is approximately 26.43 seconds.
(b) To find the distance the vehicle moved from start to stop, we need to calculate the distance covered during acceleration and deceleration separately.
For the acceleration phase:
We can use the following equation, which relates the displacement (distance traveled), initial velocity, final velocity, and time:
s = ut + 0.5at^2,
where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time taken.
Plugging in the values, we have:
s = 0 * 10.43 + 0.5 * 2.3 * (10.43)^2,
s ≈ 286.74 meters.
For the deceleration phase:
Using the same equation:
s = 24 * 16 + 0.5 * (-1.5) * (16)^2,
s ≈ 192 meters.
The total distance covered would be the sum of the distances during acceleration and deceleration:
Total distance = 286.74 + 192,
Total distance ≈ 478.74 meters.
So, the vehicle moves approximately 478.74 meters from start to stop.