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

A car is stopped at a traffic light. It then travels along a straight road such 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 $t=10.0 \mathrm{~s}$. (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?

A car is stopped at a traffic light. It then travels along a straight road such 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 $t=10.0 \mathrm{~s}$. (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

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time. It is obtained by taking the derivative of the displacement function with respect to time, reflecting how the object's position is changing at a particular instant.

Differentiation in Calculus

Differentiation is a calculus technique used to calculate rates of change. In kinematics, differentiating the displacement function yields the instantaneous velocity function, which describes the object's speed and direction at any moment.

Displacement Function

The displacement function gives the position of an object as a function of time. It allows us to determine the object's location at any specific time and forms the basis for deriving further kinematic quantities, such as velocity and acceleration.

Average Velocity

Average velocity is defined as the total displacement divided by the total elapsed time over a given interval. It provides a measure of an object's overall speed and direction over that period, smoothing out any variations in motion.

Rest Condition in Kinematics

The rest condition occurs when an object's velocity is zero. In problems involving motion, setting the velocity function equal to zero and solving for time helps determine when the object stops moving.

Transcript

00:03 So before we see, we solve the problem, we need to know what is the given condition.

00:11 The problem already gave us the equation, which is the position equation, which is x equal to 2 .4 times t squared minus 0 .12 times cubic.

00:29 So this is position equation.

00:32 Now, let's solve the, let's solve the problem, problem.

00:37 What's the part a? the part a wants to find out the average velocity between t equal 0 and the 10.

00:52 So for the definition of the velocity, average velocity, which is equal to delta t over delta x over delta t.

01:07 So what is the delta x? delta x means the position at 10 minus position at 0.

01:19 So divided by 10, the delta t equal to 10.

01:26 So at this position, we want to find out what is the x, the time at the 10 seconds.

01:36 Then, plugging the equation 2 .4 times 10 times, 10 square minus 0 .12 10 cubic.

01:53 So this will be equal to 120 meters.

02:03 So this is the position when time equals 10.

02:07 So the other one is what's the position when time equal to 0? this is the initial position.

02:15 So from the equation we understand that the position will be 0.

02:20 So, progging the definition of average velocity, so this will become to 120 minus 0 divided by 10.

02:33 So this is the 12 .0 meter per second.

02:41 So this is the answer for part a.

02:45 Now let's see what is the part b.

02:51 Part b want to find out the issue.

02:55 Instantaneous velocity at the time equal to 0, 5, and 10 seconds.

03:06 So before we solve the problem, we need to understand what is the instantaneous velocity.

03:16 The instantaneous velocity, the definition is this, the delivery of function of x position function.

03:27 So after we do the derivative, so the equation will become 4 .8 times t minus 0 .36 times square.

03:48 So this is instantaneous velocity equations...

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