A car is stopped at a traffic light. It then travels along a straight road so that its distance from

the light is given by x(t) = bt2 – ct3, where b = 2.40m/s2 and c = 0.120m/s3. (a) Calculate the average
velocity of the car for the time interval t = 0 to t = 10.0 s. (b) Calculate the instantaneous velocity of the
car at t = 0, t = 5.0 s, and t = 10.0 s. (c) How long after starting from rest is the car again at rest?

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To calculate the average velocity of the car for the time interval t = 0 to t = 10.0 s, we need to find the displacement of the car during this time interval and divide it by the time elapsed.

(a) Average velocity calculation:
1. Calculate the displacement during the time interval:
- At t = 0 s, the initial position is x(0) = b(0)^2 - c(0)^3 = 0.
- At t = 10.0 s, the final position is x(10.0) = b(10.0)^2 - c(10.0)^3.

2. Calculate the difference between the final and initial positions to find the displacement:
Displacement = x(10.0) - x(0).

3. Calculate the average velocity:
Average velocity = Displacement / Time elapsed = (x(10.0) - x(0)) / (10.0 s - 0 s).

Now we move on to calculating the instantaneous velocity at different time points.

(b) Instantaneous velocity calculation:
To find the instantaneous velocity at various time points, we need to take the derivative of the position function x(t) with respect to time t.

1. Take the derivative of x(t) = bt^2 - ct^3 with respect to t:
dx/dt = 2bt - 3ct^2.

2. Substitute the values of t into the derivative equation:
- At t = 0 s, substitute t = 0 into dx/dt equation.
- At t = 5.0 s, substitute t = 5.0 into dx/dt equation.
- At t = 10.0 s, substitute t = 10.0 into dx/dt equation.

(c) To find the time when the car is at rest after starting from rest, we need to find the time point where the velocity is zero. We can use the same derivative equation from step 1 in part (b) and set it equal to zero to solve for t. Solve the equation:

2bt - 3ct^2 = 0.

Solve for t to find the time at which the car is at rest.

By following these steps, you can find the answers to all three parts of the question.