The displacement (in feet) of a particle moving in a...

The displacement (in feet) of a particle moving in a straight line is given by $s=\frac{1}{2} t^{2}-6 t+23,$ where $t$ is measured in seconds. (a) Find the average velocity over each time interval: $\begin{array}{ll}{\text { (i) }[4,8]} & {\text { (ii) }[6,8]} \\ {\text { (iii) }[8,10]} & {\text { (iv) }[8,12]}\end{array}$ (b) Find the instantaneous velocity when $t=8$ . (c) Draw the graph of $s$ as a function of $t$ and draw the secant lines whose slopes are the average velocities in part (a). Then draw the tangent line whose slope is the instantaneous velocity in part (b).

The displacement (in feet) of a particle moving in a straight
line is given by $s=\frac{1}{2} t^{2}-6 t+23,$ where $t$ is measured in
seconds.
(a) Find the average velocity over each time interval:
$\begin{array}{ll}{\text { (i) }[4,8]} & {\text { (ii) }[6,8]} \\ {\text { (iii) }[8,10]} & {\text { (iv) }[8,12]}\end{array}$
(b) Find the instantaneous velocity when $t=8$ .
(c) Draw the graph of $s$ as a function of $t$ and draw the secant
lines whose slopes are the average velocities in part (a).
Then draw the tangent line whose slope is the instantaneous velocity in part (b).

Key Concepts

Secant Line

A secant line is a straight line that connects two points on the graph of a function. Its slope represents the average rate of change between those points, which in the context of motion is the average velocity over the time interval between those points.

Instantaneous Velocity

Instantaneous velocity is the rate of change of displacement at a particular moment in time. It is found by computing the derivative of the displacement function at that point, reflecting the slope of the tangent line to the graph at that specific time.

Tangent Line

A tangent line touches the graph of a function at just one point (without crossing through it immediately) and has the same slope as the function at that point. The slope of the tangent line represents the instantaneous velocity, as it shows the precise rate of change of displacement at that exact moment.

Displacement Function

This function represents the position of a particle along a line as a function of time. It encapsulates how the particle’s position changes and is used to analyze motion, providing a basis for understanding both average and instantaneous rates of change.

Average Velocity

Average velocity over an interval is defined as the change in displacement divided by the change in time. It is computed by taking the difference in the displacement function's values at the endpoints of the interval and dividing by the time elapsed, representing the slope of the secant line connecting the corresponding points on the graph.

Transcript

00:01 Today we will be finding the average velocity over several different time intervals using the displacement equation s equals 1 half t squared minus 6t plus 23 and also the equation the average velocity equals f of b minus f of a over b minus a.

00:15 For the first interval for 8, a will equal 4 and b will equal 8.

00:21 So we'll start with a larger one, which is 8 and we'll just plug it into the displacement.

00:26 1 half of 8 squared minus 6 times 8 plus 23 and 1 half of 8 squared is 32 minus 48 plus 23 equals 7 for the second part 1 half of 4 squared minus 6 times 4 plus 23 1 half of 4 squared is 8 minus 24 24 plus 23 equals 7.

01:00 So plug it into the equation, 7 minus 7 over 8 minus 4 equals 0 feet per second.

01:11 For the second interval, we again have 8, so 1 half of 8 squared minus 6 times 8 plus 23.

01:20 We already know 7.

01:22 For 6, we have 1 half of 6 squared minus 6 times 6 plus 23.

01:29 1 half of 6 squared is 18 minus 36 plus 23 equals 5.

01:37 So plugging that in, we have 7 minus 5 over 8 minus 6, which equals 2 over 2, and that equals 1 foot per second.

01:50 For the third one, we start off with 10 as a larger one.

01:53 So 1 half of 10 squared minus 6 times 10 plus 23.

01:59 1 half of 10 squared is 50 minus 60 plus 23 and that equals 13.

02:07 For 8, we already know is 7.

02:09 So 1 half of 8 squared minus 6 times 8 plus 23 equals 7.

02:16 And putting that in, we get 13 minus 7 over 10 minus 8, which is 6 over 2.