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An object leaves the point $(0,0,1)$ with initial velocity $mathbf{v}_{0}=2 mathbf{i}+3 mathbf{k}$. Thereafter it is subject only to the force of gravity. Find a fo...

Question

An object leaves the point $(0,0,1)$ with initial velocity $mathbf{v}_{0}=2 mathbf{i}+3 mathbf{k}$. Thereafter it is subject only to the force of gravity. Find a formula for the position of the object at any time $t>0 .$ Use feet and seconds.

An object leaves the point $(0,0,1)$ with initial velocity $mathbf{v}_{0}=2 mathbf{i}+3 mathbf{k}$. Thereafter it is subject only to the force of gravity. Find a formula for the position of the object at any time $t>0 .$ Use feet and seconds.



Answers

An object is projected upward with initiall velocity $ v_0 $ meters per second from a point $ s_0 $ meters above the ground. Show that
$$ [v(t)]^2 = v_0^2 - 19.6[s(t) - s_0] $$

So we want to show if we have an object their own. From initial height eight zero with an initial boss, the zero that the height at Time team is given by the function H NT is equal to one half a T squared plus zero t plus a syrah where ji is acceleration due to gravity which is just a constant that has value negative thirty two feet per second squared. So first, we might want to recall the relationship between some of these things. Or maybe first. What we should say is at this height function is really just another way of saying displacement. So we could think of this as our s a T function, which is how displacements often written. So recall that birds the derivative Oh, her vacation right. The lost city is equal to the derivative of our displacements. On this case will be h of tea. And if we were to integrate each of these with respect to me So the derivative and anti drivable cancel out and we'LL be left with each of you. So if we integrate our velocity function, we should get And then we also know that our acceleration function is equal to the derivative of velocity. So if we go ahead and integrate this for the interval of our acceleration and interval of our velocity give so so the derivative and consults who will be left with Kitty. So if we want to find h of T, we're going to have to take our acceleration function and integrate it to tots. So here were told that our acceleration is G which is a constant. So my first want to find B a t. You don't need to find the ti If I want to find h t. I want to integrate my acceleration function on DH I'LL just go ahead and plug this image G since in the h t here we also have it is so this will be the integral of a constant G and recall bits. This here is really teeth zero power so we can go ahead and use this cable property to rewrite this Is he into role of t zero and then interpreted. This will give G He's zero plus one divided by one constant C. And this here would simplify down to gene of tea. See looking. So now we're told that our initial velocity is zero. So that means when our time is zero, we should get B zero out. So the of zero is equal to the zero which is equal to so g of T. C and I program zero here and then this here would help me. V of zero is equal to C. So I have this here and now we go ahead and take this and put it into our velocity equation. So using that and this, we can get that the team is equal to the of Tea Time's team, plus the there. Now if I want to find my height h of team, remember, I want to integrate all this over a little bits. So now if I want to find by this placement, remember, this is equal to So I take the anti derivative of this here between anti revenue. That and I could go ahead and read this s o first using the scaler property on DH, some property of integration. So just find this key plus zero and then remember, there's an implying t zero right here. So distributing into goal Simple now. So DDT there, there. And so the sense of becoming so this is the first power. So leave power rule. So teach the one that's one over the new power which will be too put some constant of integration. Cox one wass the zero t zero lost one over the new power which one? And there was a little bit. And then plus another constant of integration Scene two. Great. So we can go ahead and simplify this a little bit and re break this one half He t square plus B zero t plus. And since we had these two constants of integration, we can just have them get a symbol constant. So we have that This here is chastity, and we're told that this has an initial height of eight zero. So we want to use that fact. So So, actually, this shouldn't be a city, but actually h of team. So it's still displacement, but just using a different variable. So now the probie h of zero is equal to H zero. And now I want to go ahead and plug in zero for all my teams won half he piece where the hero see, And then I can plug in zero here here So these first two terms go away and I'll have a check. Zero isn't privacy. Then we can go ahead and plug this in with its So combining these two packs, we end up with a tch of tea. Is it one square wass? The zero T plus ate cereal and this is exactly what we were wanting to find.

Clear. Ask to show that the following functions trip So I start off. We can find We know the M T is equal to negative 9.8 meters squared meters per second squared. And so then we know that 50 is in a derivative of a hefty which will get us nine nega 9.8 plus a constant. And so then we know that the city is anti derivative Viotti which will get us negative 4.9 t squared plus res ear of tea plus Sarah So that's our constant. Now we know that V A T squared is equal to no denying point. Puts read to the do you know squared and so we can actually factor that out to get us negative 9.8 Like the man 0.80 plus reaches zero squared is equal to maybe 9.8 t plus beaches Air It was 99.80 plus of peaches Oh, remember we're finding V A t squared from this expression over here Salt scroll down low but to give us a little bit more room. And so now what we can do is we can for you it out to get us naked. 96. Hey sakes. 0.0 for T squared minus 19.6. Reserve T plus three squared of it's p zero which will get us so we can rewrite their size. Me 20 My next 19.6 maybe 4.9 t squared plus reaches a roti. And so we know that a city is equal to what we just found earlier than before. 0.9 t skirt plus feet zero t plus us +20 And so if we rewrite it, we get a sooty. My s s zero is equal to negative. 4.9 t squared. He squared plus reaches zero teak in. So based off of that information, we can plug it right back into our original function to get us. So we have about right back into three t squid. We can replace it with this section over here, which gets us be of tea. Squared is equal to you. Squared zero minus 19.6 s o t. My next guest is there

In this brother were given that balls the be there. It's all position. Yes, over DT And we're giving him a loss of function in time for tea and were given an initial position and were asked to find the position. So we'll use the fact that derivative off come sign by t is equal. Do negative high sign t So for miss, we go backwards if you take the incredible philosophy Fine s then to be, um negative o co sign high tea Since we're missing missing this client, let's just buy this pot pie What? Some constancy. And we'll use this constant using the given commission. So we know that as people it's that t zero is equal to zero and that is a negative of co sign zero divided by five plus c considering one sort of one for seed. And from this we find C to be well, that actually is one of pine, so we might see to be negative one four. Bye. I'm sorry because I zero is equal to one and we have negative here so that I get one and we finally see to be all of their wonderful prize. If you plug this in right here, the five position function s to be won over by minus homeside t

Okay, so we know the acceleration as a function of time, which is the derivative of velocity, which is the second derivative of position is my point. Okay, that's acceleration due to gravity, actually in meters per second squared. Really? So we know that he zero is negative. Three s zero zero. So to find the we need to find an anti derivative of nine point eight last nine point eight t plus some constant scene, every plugging zero. We should get negative three displaying in zero for tea. That gives me zero plus c to see his negative three. So that tells me that the velocity is really nine twenty mystery and in the position is going to be an anti derivative of dysfunction. So that would be four point nine t squared, minus three tea, plus some other constancy crime. But if I plug in zero for ta, get zero. That serum on Syria. Less crime. So see, prime is zero. So my position function is actually just four point nine t squared minus thirty. Oh,


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