A truck on a straight road starts from rest and accelerates at 2.0 m/s2 until it reaches a speed of 20 m/s. Then the truck travels for 20 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.0 s.
(a) How long is the truck in motion?
s
(b) What is the average velocity of the truck during the motion described?
m/s
To answer these questions, we need to break down the motion of the truck into different phases. First, let's analyze the motion during each phase and then calculate the total time and average velocity.
Phase 1: Acceleration
The truck starts from rest and accelerates at a constant rate of 2.0 m/s^2 until it reaches a speed of 20 m/s. We can use the equation of motion to find the time it takes to reach this speed:
v = u + at
where:
u = initial velocity (0 m/s)
v = final velocity (20 m/s)
a = acceleration (2.0 m/s^2)
t = time taken
Plug in the values:
20 = 0 + 2.0t
Solving for t:
t = 20/2.0
t = 10 s
So, the time taken to reach the speed of 20 m/s is 10 s.
Phase 2: Constant Speed
The truck then travels for 20 s at a constant speed of 20 m/s.
Phase 3: Braking
The truck applies the brakes and stops in a uniform manner in an additional 5.0 s.
(a) Total time in motion:
To find the total time in motion, we add up the times for each phase:
Total time = time for acceleration + time at constant speed + time for braking
Total time = 10 s + 20 s + 5 s
Total time = 35 s
So, the truck is in motion for 35 seconds.
(b) Average velocity:
To find the average velocity, we use the formula:
Average velocity = total distance / total time
In the first phase, the truck accelerates, so we need to find the distance traveled during this phase. We can use the equation of motion:
v^2 = u^2 + 2as
where:
v = final velocity (20 m/s)
u = initial velocity (0 m/s)
a = acceleration (2.0 m/s^2)
s = distance traveled
Rearranging the equation, we get:
s = (v^2 - u^2) / (2a)
s = (20^2 - 0^2) / (2 * 2.0)
s = 200 m
In the second phase, the truck travels at a constant speed of 20 m/s for 20 seconds. So, the distance traveled is:
Distance = speed * time
Distance = 20 m/s * 20 s
Distance = 400 m
In the third phase, the truck comes to a stop, so the distance traveled is 0 m.
Now we can calculate the average velocity:
Average velocity = (200 m + 400 m + 0 m) / 35 s
Average velocity = 600 m / 35 s
Average velocity ≈ 17.14 m/s
So, the average velocity of the truck during the described motion is approximately 17.14 m/s.