Problem 8 Medium Difficulty

The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion $ s = 2 \sin \pi t + 3 \cos \pi t $, where $ t $ is measured in seconds.

(a) Find the average velocity during each time period:
(i) $ [1, 2] $ (ii) $ [1, 1.1] $
(iii) $ [1, 1.01] $ (iv) $ [1, 1.001] $
(b) Estimate the instantaneous velocity of the particle when $ t =1 $.

Answer

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Video Transcript

this's problem Number eight of the Stuart Calculus. In addition, section two point one the displacement in centimeters of a particle moving back and forth along a straight line is given by the equation. Emotion s equals to sign. Pi T plus three co signed by team. Where'd he is measured in seconds. Party. Find the average velocity there in each of the time periods showing up. We were called at the average velocity is associated with the slope, as in, there's a rise overrun. This is a position versus time graph. So this is a change in S o change in tea and we see that in party, each of the intervals is based around t equals one. T Hawthorne will be important to figure out what s will be. So we'LL take a guess and then solver s when he is one that gives us to sign a party plus three consent of pie and playing this into a calculator. Cas s zero plus three times negative one or negative three. And therefore at time T equals one. The position is negative three centimeters and that's our reference. And what we do is we're going to find the difference in the position as final minus this initial and the numerator divided by time, final minus time, initial and the denominator and the time initial will be t equals One position initial will be negative. Three. And so we proceeded to to our calculations at the given time. Intervals, Um, the first intervals from one to two. So the time the final time is two seconds. The equation of motion provided gives this hour yes. Ah, value. And this is again something you would just plug into your calculator to figure out two time sign of pi. Times two plus three times. Co centaur pi times to thank you's This S is three. This is our tell Tess. It is three minus negative three. Which is the initial value we found here. Delta team is thie. Given time T equals two minus one. This is basically the interval length, and then average velocity is thie quotient of both. So six to eight by one gives us a slope or an average velocity of six. Repeating for the next Tim Mitchell from one to one point one. We get this for the value Bess. He evaluated at one point one which gives us a doctor s and a Delta team and a subsequent slope of about negative of four point seven one two. For the next ten interval, we repeat the same steps, get a soap of negative six point one three four approximately And finally, for the last central from one teakwood wanto t equals one point zero zero one. We can't that he stop his negative six point two six eight three seven Oh five, seven, seven Exactly. And that will be our average velocities for party purposes. To estimate the instantaneous velocity of the particle won t equals one. And since we're estimating we're going to use our spreadsheet and just continue a few more calculations because as we approach t equals on one, we approach the instantaneous velocity here for this will be the instantaneous velocity. Once this gets close enough to one, we see that the average velocity is approaching negative six point two eight three. And that's if you do the calculations enough times. So we will write that are instantaneous velocity. At one. It's negative six point two eight three centimeters per second, and that is their answer for party