A truck on a straight road starts from rest, accelerating at 2.00 m/s2 until it reaches a speed of 32.0 m/s. Then the truck travels for 76.0 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.00 s. What is the average velocity of the truck for the motion described?
To find the average velocity of the truck, we need to first find the total displacement and the total time taken for the entire motion.
Let's break down the motion into three parts:
1. Acceleration phase: The truck starts from rest and accelerates at a constant rate of 2.00 m/s^2 until it reaches a speed of 32.0 m/s. We need to find the displacement and time taken during this phase.
Using the equation of motion:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.
For this phase:
u = 0 m/s (truck starts from rest)
v = 32.0 m/s
a = 2.00 m/s^2
Rearranging the equation:
t = (v - u) / a
Substituting the values:
t = (32.0 m/s - 0 m/s) / 2.00 m/s^2 = 16.0 s
To find the displacement during this phase, we can use the equation:
s = ut + (1/2)at^2
Substituting the values:
s = 0 m/s * 16.0 s + (1/2) * 2.00 m/s^2 * (16.0 s)^2 = 256.0 m
2. Constant speed phase: The truck travels for 76.0 s at a constant speed. During this phase, the velocity remains constant, so the displacement is simply the product of the speed and time.
s = v * t = 32.0 m/s * 76.0 s = 2432.0 m
3. Braking phase: The truck decelerates uniformly until it comes to a stop. The time taken for this phase is given as 5.00 s. Since it comes to rest, the final velocity is 0 m/s.
Using the equation of motion:
v = u + at
For this phase:
u = 32.0 m/s (final velocity from the previous phase)
v = 0 m/s
t = 5.00 s
Rearranging the equation:
a = (v - u) / t
Substituting the values:
a = (0 m/s - 32.0 m/s) / 5.00 s = -6.4 m/s^2 (negative because it's decelerating)
To find the displacement during this phase, we can use the equation:
s = ut + (1/2)at^2
Substituting the values:
s = 32.0 m/s * 5.00 s + (1/2) * (-6.4 m/s^2) * (5.00 s)^2 = -80.0 m (negative because displacement is in the opposite direction)
Now, we can find the total displacement by summing up the displacements during each phase:
Total displacement = 256.0 m + 2432.0 m - 80.0 m = 2608.0 m
The total time taken for the entire motion is the sum of the times taken in each phase:
Total time = 16.0 s + 76.0 s + 5.00 s = 97.0 s
Finally, we can calculate the average velocity by dividing the total displacement by the total time:
Average velocity = Total displacement / Total time
Average velocity = 2608.0 m / 97.0 s = 26.88 m/s (rounded to two decimal places)
Therefore, the average velocity of the truck for the described motion is 26.88 m/s.