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 - 6t + 23 $, where $ t $ is measured in seconds. (a) Find the average velocity over each time interval: (i) $ [4, 8] $ (ii) $ [6, 8] $ (iii) $ [8, 10] $ (iv) $ [8, 12] $ (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 - 6t + 23 $, where $ t $ is measured in seconds.

(a) Find the average velocity over each time interval:
(i) $ [4, 8] $ (ii) $ [6, 8] $
(iii) $ [8, 10] $ (iv) $ [8, 12] $

(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).

Transcript

00:01 In this activity, we're given that the displacement of a particle moving in a line is given by this function, 1 .5t squared minus 6t plus 23, where t is time in seconds.

00:15 And we're being asked to find the average velocity at several different intervals, over several different intervals of time, first from four seconds to eight seconds.

00:30 So the average velocity of this particle between four seconds and eight seconds.

00:35 Well, average velocity is given to me as the change in displacement divided by the change in time.

00:58 So i want to calculate the change in time, and i want to calculate the displacement at the two different times, and subtract them to get the change in displacement at each time.

01:16 And then that will give me my average velocity.

01:36 So i'm going to set this up so that i can calculate displacement.

01:41 The average velocity between 4 and 8, let me just draw this instead of typing it, the average displacement between 4 and 8 is going to be the displacement at 8.

01:55 So when time equals 8, displacement equals 5, 7, minus the displacement at 4 seconds.

02:13 The displacement at 4 seconds also equals 7 divided by the change in time, which is 8 minus 4.

02:24 7 minus 7 is 0.

02:27 8 minus 4 is 4 .0 divided by 4 is 0.

02:30 So the displacement between 4 seconds and 8 seconds is 0.

02:35 Then the displacement between 6 seconds and 8 seconds is going to be given to me.

02:45 The average velocity will be given to me by the displacement at 8 seconds, which we already know is 7, minus the displacement at 6 seconds.

02:57 The displacement at 6 seconds, oops, is 5 over the change in 10.

03:08 Which is six seconds minus four seconds.

03:12 Seven minus five is two.

03:15 Six minus four is two.

03:18 Two divided by two is one.

03:20 So the average velocity over that time interval is one.

03:25 Then the time interval, eight seconds to ten seconds, velocity is given to me by the change in time, or the change in displacement divided by the change in time.

03:42 So the displacement of this particle at 10 seconds is 13.

03:52 And this displacement at 8 seconds, we already know, is 7 divided by 10 minus 8.

04:01 13 minus 7 is 6.

04:05 10 minus 8 is 2.

04:08 6 minus 2 is 3.

04:10 So the average of velocity is 3 feet per second over that interval.

04:14 And then finally, we're asked to calculate the displacement from 8 to 12 seconds.

04:29 So the average velocity is going to be the displacement at 12 seconds, which is 23 minus the displacement at 8 seconds, which is 7, divided by 12 minus 7.

04:55 23 minus 7 is 16, 12 minus 8 is 4.

05:01 16 divided by 4 is 4.

05:09 So the average velocity between 8 seconds and 12 seconds is 4 feet per second.

05:17 That's our part a.

05:24 For part b, we are being asked to find the instantaneous velocity at time equals 8 seconds.

05:38 So i am going to go ahead and erase this so that i've got room to work on that.

05:52 So we remember that for part a, the velocity is part one was zero.

06:03 The velocity on the second time interval was one.

06:08 The velocity on third time interval was three.

06:14 The velocity on the fourth interval was four.

06:22 For part b, the instantaneous velocity is given to me as the limit of the average velocity as some time gets closer and closer to the input that i'm interested in eight.

06:45 So my input time is getting closer to eight seconds, and i am finding what is the limit of the average velocity as the difference between the time and eight seconds itself gets smaller and smaller.

06:59 So that is the displacement at time t minus the displacement at time eight divided by t minus eight...

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