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Problem 7 Medium Difficulty

The table shows the position of a motorcyclist after accelerating from rest.

(a) Find the average velocity for each time period:
(i) $ [2, 4] $ (ii) $ [3, 4] $ (iii) $ [4, 5] $ (iv) $ [4, 6] $
(b) Use the graph of $ s $ as a function of $ t $ to estimate the instantaneous velocity when $ t = 3 $.

Answer

A.(i) $29.3 \mathrm{ft} / \mathrm{s}$
(ii) $32.7 \mathrm{ft} / \mathrm{s}$
(iii) $45.6 \mathrm{ft} / \mathrm{s}$
(iv) $48.75 \mathrm{ft} / \mathrm{s}$
B. $29.7 \mathrm{ft} / \mathrm{s}$

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Video Transcript

This is problem number seven. Ah, See, Stuart Calculus, A petition section two point one. The table shows the position of a motorcyclist after accelerating from rest. So we look into the book. Here's the table that they're referencing. This table shows the position, um, in feet. So the position ass and feet and the time associated with each of the positions parties to find the average velocity for each time period came. Let's recall that average velocity can be thought of as a slope where you have a rise divided by a run. In this case, the rise has to do with a change in position, which is denoted by S in this case from the table over time and again. We know this because we generally see position versus time as an escort does t graft. Maybe it why versus t And so a slope will be rise, which is a change in this overrun, which is a change and tea. So average velocity is Delta s over Delta T and we'LL see that from the table. We're given the s values, Eddie to these time periods two seconds, three seconds, four seconds, Five seconds in six seconds so we just We need to choose the final position. Subject the initial position from that and also for the denominator. Take the final time, subtract the initial time and that gets us our average velocity. Hey, so let's go to our spreadsheet, where we have translated the table directly from the book here T seconds as feet. And here each of the solutions for party for the interval, two to four seconds. Notice that we do. The position s at four seconds. Seventy nine point two minus the position as at two seconds. Twenty point six. And the numerator that's dealt ass divided by the change in time. Four seconds. When is two seconds and that is got to in the developed later. And finally, the average velocity press in feet per second is twenty nine point three for part one of party part two. For the into rule three to four seconds, the average velocities thirty two point seven feet per second. Part three. The average velocity is forty five point six feet per second for the interval, forty five seconds and for part for a party from Portis. Six seconds for that time interval. The average velocity is forty eight point seven five feet per second and that is party our party. We're going to use the graph of PS function versus team to estimate the instantaneous velocity when t equals three. So we're going to do this by estimating a tension line again. Begin the X axis. This blue line represents the S as a function of team. And here we have drawn a line that's tangent at T cool story. And the slope of this line is our average velocity AT T go three seconds and this line is specifically drawn and his tension because it only touches the s function. At one point here Atika Stream and we also do it so that we see the height that it reaches and the point that it reaches here. So it reaches a height of one twenty at five point five seconds. And here the line intersex zero feet. At one point six we see here there is a difference. Twenty eight minus zero. That's the rise. Goodbye. A final time by point five minus the initial time. One point five. Again, this is to hope. Estimate the slope of this tension line or the average velocity and get an estimate of approximately thirty per second. Since we did this, it's an estimate thirty feet per second. Ah, will be Hower Final answer for party