A web page designer creates an animation in which a dot on a...

A web page designer creates an animation in which a dot on a computer screen has position $$\vec{r}=\left[4.4 \mathrm{~cm}+\left(2.8 \mathrm{~cm} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\imath}+(5.5 \mathrm{~cm} / \mathrm{s}) t \hat{\jmath}$$ (a) Find the magnitude and direction of the dot's average velocity between $t=0$ and $t=2.0 \mathrm{~s}$. (b) Find the magnitude and direction of the instantaneous velocity at $t=0, t=1.0 \mathrm{~s}$, and $t=2.0 \mathrm{~s}$. (c) Sketch the dot's trajectory from $t=0$ to $t=2.0 \mathrm{~s}$, and show the velocities calculated in part $(b)$.

A web page designer creates an animation in which a dot on a computer screen has position
$$\vec{r}=\left[4.4 \mathrm{~cm}+\left(2.8 \mathrm{~cm} / \mathrm{s}^{2}\right) t^{2}\right] \hat{\imath}+(5.5 \mathrm{~cm} / \mathrm{s}) t \hat{\jmath}$$
(a) Find the magnitude and direction of the dot's average velocity between $t=0$ and $t=2.0 \mathrm{~s}$. (b) Find the magnitude and direction of the instantaneous velocity at $t=0, t=1.0 \mathrm{~s}$, and $t=2.0 \mathrm{~s}$.
(c) Sketch the dot's trajectory from $t=0$ to $t=2.0 \mathrm{~s}$, and show the velocities calculated in part $(b)$.

Key Concepts

Position Vector

The position vector describes the location of an object in space at any given time, using components along defined coordinate axes. It encapsulates both magnitude and direction to clearly specify where the object is relative to an origin.

Average Velocity

Average velocity is defined as the total displacement divided by the total time interval. It provides a vector quantity that summarizes the overall change in position over a specified duration, indicating both how quickly and in what direction the object has moved.

Instantaneous Velocity

Instantaneous velocity represents the rate of change of the position vector at a specific moment in time. It is obtained by differentiating the position vector with respect to time, yielding a real-time measure of the object's speed and direction.

Trajectory

The trajectory of an object is the path it follows through space, which can be traced out by plotting the successive positions given by the position vector. Understanding the trajectory helps in visualizing the overall motion pattern of the object over time.

Transcript

00:01 Okay, we have a dot on a screen which follows this position equation as a function of time.

00:08 So we see in the ihat direction it follows this, so 4 .4 plus 2 .8 times t squared.

00:17 And then the y direction, it follows this 5 .5 times t.

00:23 And what we want to find out is the average velocity.

00:30 During from t equals 0 to 2 seconds.

00:33 So we know that average velocity looks like this.

00:40 So for the x component of our velocity, it's going to be the change in position, divided by the change in time.

00:47 And then vy, of course, it's going to be the same thing.

00:50 It's not a vector.

00:51 It's average y velocity, the change in y position, change in time.

00:58 And so what we need is the initial position and final position.

01:03 So the initial position, we get just by plugging t equals 0 in.

01:10 So that's going to be 4 .4 centimeters in the i -hatt direction, an x direction.

01:19 And then the final we get for plugging in t -cels 2.

01:24 So for that, we get 15 .6 centimeters, and then 11 centimeters in the y direction.

01:35 All right.

01:36 And now we have these so we could calculate, what are we.

01:38 Our average velocities are.

01:40 So average vx velocity is going to be our final position, 15 .6 centimeters minus 4 .4 centimeters, divided by the change of time, which is 2.

01:57 And what we get from this is 5 .6 centimeters per second.

02:05 And then vy is quite similar, which is going to be 11 over 2.

02:11 Or 5 .5 centimeters per second.

02:15 So this is the average velocity and since we wrote it component -wise, this is the vector of average velocity, which we know both contains magnitude and direction.

02:28 Okay, so let's move on to part b.

02:34 So for part b, we want to know the instantaneous velocity...

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