CALC A car is stopped at a traffic light. It then travels...

CALC 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 $x(t)=b t^{2}-c t^{3},$ where $b=2.40 \mathrm{m} / \mathrm{s}^{2}$ and $c=0.120 \mathrm{m} / \mathrm{s}^{3} .$ (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 $t=0, t=5.0 \mathrm{s},$ and $t=10.0 \mathrm{s}$ . (c) How long after starting from rest is the car again at rest?

Key Concepts

Root Finding for Zero Velocity

Determining when an object is at rest involves solving for the time at which the instantaneous velocity is zero. This requires setting the derivative of the position function equal to zero and finding the non-trivial solution that represents the moment the object returns to rest.

Average Velocity

Average velocity is computed as the total displacement divided by the total time interval. It provides an overall measure of the object's motion over a period, even when the speed is not constant.

Position Function

The position function specifies the displacement of an object as a function of time along a straight path. It is the cornerstone in kinematics for describing the location of an object at any given moment.

Differentiation

Differentiation is the process of computing the derivative of a function, which measures the rate of change of a quantity. In kinematics, differentiating the position function gives the instantaneous velocity and, if differentiated further, the acceleration.

Instantaneous Velocity

Instantaneous velocity is the derivative of the position function with respect to time and represents the object's speed and direction at a specific moment. It is essential for understanding the object's behavior at any particular point in time.

Transcript

00:01 We're given a relationship for position x as a function of t, so x of t, and it's equal to b times t squared minus c times t to the third, where b and c are both constants, and i have those written out here.

00:13 So for part a of our question, we are asked to find the average velocity of the car in the time interval between t equals zero and t equals 10 seconds.

00:24 So to find this velocity, v, we need to know the position at time t equals zero, and the position, at time t equals 10 seconds.

00:33 So the position at time t equals 0 is x of 0.

00:36 So we're just plugging 0 into that expression.

00:38 But b times 0 squared minus c times 0 to the third is just 0.

00:43 So 0 meters is the position at x equals 0.

00:46 Now how about x equals 10? so we simply plug in 10 into our equation.

00:50 So it would be the constant b times 10 squared minus the constant c times 10 to the third.

00:55 And we find that this is equal to 120 meters.

01:01 So now to find the average velocity, since velocity is distance divided by time, we simply take the distance, which would be x of 10 minus x of 0, and we divide it by the total time, which would be 10 minus 0 seconds, or in other words, divided by 10 seconds.

01:19 So plugging in those values, we find that the velocity here on average is 12 meters per second.

01:28 You can box it in as a solution for a.

01:31 Part b wants us to find the instantaneous velocity at time t equals 0, at time t equals 5 and at time t equals 10...

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