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A) For the following height values, compute the quadratic of best fit, h-atl+bttc for an object launched upwards at t=0.2)(seconds) 0.5 h (meters) 502.0 1253.5 1505...

Question

A) For the following height values, compute the quadratic of best fit, h-atl+bttc for an object launched upwards at t=0.2)(seconds) 0.5 h (meters) 502.0 1253.5 1505.0 1356.5 70b) Based on your model, compute the initial height, the maximum height, and the time when the object hits the ground:c) Compute the least squares error and identify the point which fits the model the worst.d) Sketch the graph of the quadratic and the original data values on the same graph: Include the least square errors i

a) For the following height values, compute the quadratic of best fit, h-atl+bttc for an object launched upwards at t=0. 2) (seconds) 0.5 h (meters) 50 2.0 125 3.5 150 5.0 135 6.5 70 b) Based on your model, compute the initial height, the maximum height, and the time when the object hits the ground: c) Compute the least squares error and identify the point which fits the model the worst. d) Sketch the graph of the quadratic and the original data values on the same graph: Include the least square errors in your sketch:



Answers

A ball is thrown upward, and its height, $h,$ in metres above the ground after $t$ seconds is given by $h(t)=-5 t^{2}+25 t, t \geq 0$ a. Calculate the ball's initial velocity. b. Calculate its maximum height. c. When does the ball strike the ground, and what is its velocity at this time?

So for this we have h of t equals 32 T minus 16 T Square. Let's go ahead and write this in the form of her Abdullah. So we take negative 16 out common. That leaves us with tea square minus two t. And then if we add a one to complete the square, we've effectively subtracted sixteens. We need toe, add 16. So then are probably equation simplifies to negative 16 times T minus one whole square plus 16. So that gives us off Vertex or a maximum height off one comma 16. So that tells us one second after the after the boldest thrown, it reaches its maximum height of 16 feet for the second part we need to solve for when the ball hits hits the ground again. This implies a true T equals zero that's equal to 32 t minus 16 T square, and we need to solve for T a free. So for TV, take 16 t out common and have start negative 16 t up common and have t minus two there, so that leaves us with a T equals zero and T equals two. So this tells us that the ball is going to be background level AT T equals zero seconds or right before it's thrown or after two seconds. That means it's gonna hit its maximum point after a second, and then it's gonna take another second to come back to the ground.

Hello. In this bowler number 57 it is told that person is standing close to the edge off a top of her 1 60 foot building through the baseball vertically upwards. OK, something is given. It's a question is given Esty. Yes. T minus 16 DS squired. Bless 64 b bless 1 60 which is well in the farm off a X Esquire. Blessed be x bless. See Okay with the potential parabola. So go to represent parable as to in friggin So our problem is question number A After how many seconds is the boiled ages? The maximum here, if become bad is listing zero because it is equal to minus 16 so there will be minimal. Also there will be Maxima and vortex will be like this. Okay, No, for this case for this case, the Vertex are in our according to our give any question vortex will be de Goma s de That is he will be in place off x an SD And what this tea will be. He will be minus B over the way that if he compared this be good minus 64 over doing to minus 16. That is minus 64 over my minister Tito. That is, too. It means when d equal to two seconds in the ball bill born real reach its maximum value. It's maximum height, it's maximum height. And what is that message? Might this maximum height will be value off as the value of s at he called you to which will be if we can find just plugging in equals toe in our given equation that is 64 in tow to blessed 1 60 which will be to its wherefore that in minus 64. Bless 64 into 2128 miss 1 60 that is it will be 1 60 plus 64. This is 1 63 1 60 less 64 that is 24 feet What you're thinking for. So we have maximum at time maximum at time Sequel to to second and maximum Hi equal to to 24 for so this should be the answer apart. Okay. To a second and 24 feet Maar for part B, it is told that how many seconds that it take until the ball finally here is the ground around or there nears 10 top a second it means how many seconds after eating. First off all, let us recall what will happen when ball Bill hit ball will hit the ground. Well, then, Esti become zero that is minus extend T Esquire plus 64 day bliss 1 60 equal to zero that if you take minus 16 common, it will become Ah do you square minus 40 10 equal to zero. So if you divide both sides by minus 16 Vic it Dear Square minus 40. Let's stand equal to Zito. So according Toa are quality formula You get equal to minus minus four Bless minus mine's 40 square for into one and to then Oh yes, it should be mine because we already had taken minus the common welcome minus B minds my four plus minus squares. This should be oh minus 10. Well, we're two into one, so it will be full Less minus 16 plus 40 over to that is four plus minus 56. A little 34 Bless minus or we can take for over here. Oh, no, We should take to outside and inside. It will be 14 over to No, you know, if you take two as common it will become two plus miners on little 14 over tow. This too. This too gets cancel out to plus minus on road 14. No, we should write under 14 equal to under sort Any pinto? 8.7416 find. So it becomes three, quien 74165 So we have two values off T. That is two minus three point 74165 and two plus 3.74165 Now one creatively say that this is negative. And what time cannot be negative. So this will lose neglected. So we're left with only one value. That is equal to five point 74165 And we are being told toe around to the ah, near stand off a second. So be will approximately be five point seven second. This should be dance studio Father. Second part. Okay. Malcolm's turned black. That is C to see Park s zero. Describe what days? Men s zero That is value off s de at equal to zero. So it becomes after plugging in the value in Givens equation 16 into zero square 64 in 20 bliss 1 60 that is zero plus zero. It becomes 1 60 So we have as zero equal to 1 60 foot 1 60 feet. Okay. Which means it means it demands the height off the ball. It maize. Hi. Off the bar from the ground just before throwing. Okay, which is the high initial head? That is the height of the building. Okay, the you is just one part of it. And see to craft the quality function, begin the gravity zero and inevitability borders the ground. Okay, so let us try and draw the graph. This is Esti. This is D. Well, first of all, at equal to zero, we have as two and 60 feet. Let us right here. 1 60 Here. This is a starting point. Okay? I have maximum height. That is as equal to a two second. It goes to 2.2 24. Equal toe to second, where it is registration to second. It goes to later today. 2 24 So this is the point. We should be out of our techs to 2 24 to second. The height is 24 feet. Okay. After 5.7, or at 5.7 seconds. It reached the ground. That is, I had become zero. So let us take 51 seven, second over here by 0.7 seconds. It hits the ground. So our graph, man, roughly look like this. Okay, this should be the vortex, and I mean maximum value. Okay. And again, just take it or not. Okay. The graph real roughly. You look like this.

We have given a person is standing close toe edge over stop up 160 food buildings say this is the top off the 160 ft willing and personally standing here truth a baseball, particularly aboard. So he truth a baseball vertically, upwards and after some time it will come to the ground the heretics Montagne Ste So here is the card edict, Johnson esti that represent the motion off the gold with respect to time that this minus 60 Antaeus where plus 60 food tea plus 160. So this is a question of the ball with respect to bank. And now, after how many seconds does the ball reach its make some height. That means we want expections value. And here it is. Less than zero date means vortex will give the Maxima. So here, competitive with excess where plus be explosive and equals minus 60 b equals to 64 seek was 260. And now we know that what takes well here is minus video, every which to it and f all minus be divided with to it. So now first we have to find my next video every with two way for delay. You so it will be minus very, very toe minus three. They didn't minus 64. They have it to where they're going to murder. Deployed with minus 16. This is minor 16 and it will give here to that Miss minus video every two ways to. And now we won't find f off to sign up. Or do I look 16 multiplied with two x squared plus 64 multiplied with toe plus 160 that will give here 2 to 4 tweet. That means it will come to the maximum height off six. Here. Photo 4 ft. So this is a make some height that is to do for pit and after how many times so after two seconds makes a light is 12 to 4 ft And now for the second part, this is the first part. And now for the second part, how many seconds doesn't take until the ball finally hits the ground? So a round to the nearest end of the second. So, for this question in here, that is s de equals toe minus 16 piece grip plus 64 b bless her in and 60. And now this is a question that more of the ball side after basic, its above the ground. So for this, if we want I am taking until the ball finally hits the down that when it comes to the ground equity to become zero and minus 16 80 square plus 64 b plus 160 Now take make some common. So this will weigh minus T square plus 40 plus 10 equals to this is se zero and now here equals minus one. B equals toe full and Sequels to 10. And we know that the value off the hear T equals toe. This is minus B plus minus B squared, minus four a c. They were headed with toe and now put the value so minus with the did equals two minus food plus and minus B squared, minus four a. C. That means this is 16. We square that it's 16 minus four. Does he? So this will be minor form. Will replied with this is Casey for little Way 40. So this is 40 and they were giving away. So they're dead minus two and b equals minus four plus and minus So this will be 56 divided with minus two. And now when we calculate we get P equals toe, this is 5.742 Why, Who? And seven for two. And now when we corrected, we get to the tens place. So this is 5.7 Siggins. After 5.7 seconds, it comes to the ground. And now this is the second part. And now for the sea part Fine s zero. So s Didato and describe What does this mean, Zito? That means here P equals 20 at men's initial point initial point. So put here s zero that wins here Piers equals too little. The question is, esti equals toe minus 16 80 square plus 60 40 plus 160. So he had estate that Now we have to replace with zero. So this will be Escondido. So this is little plus little plus 160. That means as little equals 260 that went ash initial Floyd, it waas 160 ft above the ground. This is the meaning. Go as little. And now we have toe use religion to grab the court etiquette question so now and begin with that equals 20 airplane. So now here is the graph. So now this is the graph and people attending the question SD equals two minus 16 D square plus 64 p plus 160. So this is the question. And either the point say this is the first point that zero and 160 zero and 160 this is a peak. Most point that is you said this is active Zika and to for and no here in the point often how many seconds it reaches to the ground. So this is the reality that 5.74 to 0. So this is a point, ABC. I'm naming it. This is the point. A on this is the point bay and this is the point, and this exist at the present Time X is, and this is the high Texas. So now we have re presenting the equation and next graph also

Okay. An object falls from H. It's gonna start at H. Uh And the initial is zero because it's starting from rest, it travels zero point 50 H. In the last one second. Okay. Time and height that it falls. Okay so um V squared equals the initial squared minus two G. Delta Y. The delta. Why? Here is going to be H -0.5 H. for the beginning of the fall. Um So V by the way V0 is zero is going to equal the square root of two G. Times what H -0.58. Which is just going to be 0.58. I'm going to call that one half h. Okay. Simplifying uh V is going to be the square root of G. H. And this will be V. So I'm gonna call this v. Sub two because it's at um halfway down. Okay. But we want the time and height of the total fall. Um The total fall. Let me think about this again. It travels that far In the last 1/2. Okay so in the last half of a second because this was the first half of a second. Now let's do the last half of a second. Um V Final which would be like I'm gonna call that three squared equals. But wait I know time here and I know height. So I'm going to use both the initial squared. Now everything is in the downward direction. Mhm. But we square it so then the velocities But I know that v. three is going to be larger than v. two. So I need to add two G. And then again it's gonna be 1/2 of H. So V. three is going to be the square root of v. two squared plus G. H. So now Um I've got v. two V. Three and H. That are unknowns. Okay let's keep going. Um Why equals why initial? God yeah let's try this for the first half. Why equals why initial should be H. Plus the initial T. Will the initial would be zero minus one half G. T. Squared. But that would just be T. For the first half. I'm going to write you know what I'm going to write I'm going to write half of the total time. So t. Is going to be the total time. Okay um And why is going to be one half age? Because it's gonna fall one half age? Okay so from this one H minus one half ages. One half H. Those cancel out. Okay so H. Over G. Is going to equal T squared over four. And so T. Is gonna equal to square root of H. Over G. So T. Equals to square root of H. Over G. I didn't put H into my calculator yet. Oh I don't know what H. Is E. E. Okay so I can't use a calculator yet. So now I have three equations but I don't know t don't know H. I don't know V. two or v. three. So I think I can do the same thing but for the bottom half So again H over two is going to equal so why equals why initial wait a minute for the last half? Why final is zero is going to be why initial minus V. Two T. Um But this is not gonna be the total T. This is going to be half of the total T. For the second half. No that doesn't make sense Gar because half the height is not gonna be half the time. Ooh have the hate not gonna be half the time. Uh huh. Mhm. Whoops. Okay so half the height is not half the time so that has to go away. That has to go away. Um This is the time minus one though. So I could write the total time is The First Time -1. So I need to write T -1 here. Okay so now get all of this out of there. Um Okay I still have a choice over G. But now it's going to be uh T -1 squared. Okay because the because the one right up here um All right so I take the square root that's just gonna be t minus one. So this is going to be the square root of H. Over G minus one plus one. Okay over here then uh Why equals Y. Initial minus V. T. But we know t. Is one right here minus one half G. And again he is one squared. Okay so this tells me age equals V. Plus G. Over to Times two. Let's take this a little bit slower. H over two is gonna be v. two plus G. Over two. Then we multiply by two. H. Is going to be two V. Two plus G. Okay so H. V. Two T. And V. One are the unknowns and I've got four equations 4 equations and four unknowns. I can solve it. Okay now um now no no no no no no now um I need time and height so um I think maybe it's gonna be easier to solve this for h. Let's do that square both sides. T squared minus two T. Plus one. H. Over G. If I multiply by G. On both sides I get h Not thrilled about this. Um T squared minus two T. Plus one. Okay got it. Mhm. Actually a lot of work. Well I guess I'll put these two together and all right G. Times T squared minus two T. Plus one equals two. V. Two plus G. Okay But I don't know v. two But V two is the square root of G. H. Okay. Um interesting. So now T squared minus two T. Plus one is going to be two square root of H over G. Plus one. The plus ones cancel out. Um So now I'm gonna get t squared minus two T minus two square root of H. Over G equals zero. Okay now I could use the quadratic formula T. Equals negative B. Which is two plus or minus the square root of B squared which would be two squared which is four minus four A. is one c. is negative two square root of H. Over G. Over to. Okay well I'm going to factor out a two From everything and then cancel out that too in the denominator and that's going to be one plus or minus the square root of one plus two square root of H. Over G. Now um T cannot be negative. So that's the answer to see the physically unacceptable solution is the negative T. Um So t is only going to be the positive solution. Okay But darn it. I still don't know h what did I just do that? I I don't know H. Um um um um Made it all the way down here. I wrote that I rode that and then V two is a square root of G. H. Oh man. Oh man how am I gonna find H. two v. 2 over G. So then I could get it in terms of v. two. This is really complicated because there's four equations and four unknowns except in these three equations, this one and this one and this one, V. three isn't even in there. So I could if I could just solve these three equations, I would be good. Um Okay well uh yeah I mean this is valid too but I got this one by putting all three equations together and so I think that I can just take this one which has H. And G. In it and I can figure out what H is. And I don't think that's going to be redundant because I put all three of them together. So um I'm gonna solve for H but then that's going to give me another T in here. E H got H and V. Two here, I've got HMT here and I've got H and V. To hear asian V two here, H&V. two here. I should put these two together and figure out what H. Is. Yeah I could do that. V two equals square root of G. H. So V. Two so H minus no equals to square root of G. H. Uh plus G. Putting those two together V. Two. No what did I just do? Oh no h equals two V. Two plus G. Okay so now I can figure out what H. Is because oh my goodness uh Oh my goodness But I need the square root of eight here because this is going to give me H -2 sq Root of G. H minus G equals zero. Now that you equal the square E. Of age then I get you squared minus two Square roots of G. You minus G equals zero. So again I can use the quadratic formula you equals two square roots of G plus or minus the square root of four. G b squared minus four A c a c four G b squared minus for a c Over two. So you is going to be two square roots of G. Again it's gonna have to B plus The Square Root of eight G. Wait a minute. Let me cancel out A two here before I simplify this 2222. Okay, so it's the square root of G Plus the Square Root of two G. Square word of G plus the square root of two G. And again this has to be positive. So uh square to G plus Square to two G. Okay, But then you is the square root of age. So the square root of H is the square root of G Plus the Square Root of two G. Square that. And I get H. Is the square root of G plus the square root of two G squared. Well at least I can put that in a calculator now. Um square root of G. A Square Root of two G. Squared. And that's gonna be a church. So H would be 57.2. Now we need t. But I've got T down here so I'll just put that in a calculator. T. Is going to equal one plus the square root of one plus 2 square root Of H over G. So that's going to be 3.41 seconds. Those answers are both reasonable answers at least. And I've already explained physically acceptable and unacceptable. T. Has to be positive and H also has to be positive. And so that's it. Those are the answer's assuming I didn't make any computational mistakes up here. And you might notice that I'm not even sure I used this equation up here. This one. I don't think I did. That's interesting. All right. Nevertheless thank you for watching


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