Problem 5 Medium Difficulty

If a ball is thrown into the air with a velocity of $ 40 ft/s $, its height in feet $ t $ seconds later is given by $ y = 40t - 16t^2 $.

(a) Find the average velocity for the time period beginning when $ t = 2 $ and lasting
(i) 0.5 seconds (ii) 0.1 seconds
(iii) 0.05 seconds (iv) 0.01 seconds
(b) Estimate the instantaneous velocity when $ t = 2 $.

Answer

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Video Transcript

Mrs Problem number five of Stuart eighth Edition, Section two point one and problem or five. Seven. If a ball is thrown into the air with the velocity of forty feet per second, its height in feet T seconds later, it's given by y equals forty team minus sixteen. T squared party. Find the average velocity or the time period. Beginning twenty equals two and lasting zero point five seconds. Is your point one seconds? They're quite zero five seconds and their point zero one seconds and we should begin with. Is that the fact that an average velocity is very similar to a slope? There's a rising there's around. Ah, there's a change in why, in this case, why his height? I did what I had to change an expert. A change in time because we're dealing mostly with a projectile, which is a wireless is time graph. This is why our virtuosity, which is a changing why we're changing time, is a slope. I mean, like, like this. It's a slip in a one versus T graph. Um, and so if we think of it that way, we should be able to calculate the soap for giving information And the first thing that we should do is establish where the aliens at T equals two seconds, and we do is we Plenty equals two into this equation already. Times two baby minus sixteen times two squared two squared is for we see that this is sixteen feet. So that is the initial location of the ball at two seconds. And for us to find a change in height for a given time after two, we have to calculate using this equation where it is. Why using them subsequent time. So, for example, the first time would be point five seconds after two or two point five seconds that we plug into this equation to get the wine and then take sixteen away from that. Why sixteen of the original position and it gives us our change in y. Our change of tea is always going to be the difference in the time after two. So each of these numbers will be adult ity, and in that way we should be able to find the average of velocity. So here we have a summary for each of these four changes in time. Pick any one. They're point five. We have two seconds. In order for us to use this equation to find where the location of the wall is, take the difference with sixteen, which is the initial position. And then here is when we take this quotient of the delta y the rise divided by the Delta team, which is a run to estimate our slip or average velocity. If we repeat this with everyone as the next time period we get and I was lost between five point six the following results at time Delta T there one five your point, their height and they open it. And finally point zero one second after t equals two. The average velocities needed twenty four one six symbols of their answers. For a party, these are the average velocities that we calculate. We can expand to this these calculations to estimate the instantaneous velocity twenty equals two. And that's report. So we take the same approach and continue decreasing the daughter. I'm making it smaller and smaller so that we can best us to meet instantaneous paucity. So as this gets closer to zero, we see that the wine you gets closer of the time gets closer to the White House gets called up to sixteen down twenty, it's closer to zero. And finally, the average velocity is much closer to negative twenty four. So our best estimate, our best estimate AT T equals two seconds is that the average velocity isn't negative twenty four feet per second.