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Point) body of mass kg is projected vertically upward with an initial velocity 31 meters per second.We assume that the forces acting on the body are the force of gr...

Question

Point) body of mass kg is projected vertically upward with an initial velocity 31 meters per second.We assume that the forces acting on the body are the force of gravity and retarding force of air resistance with direction opposite to the direction of motion and with magnitude clv(t)| where =0.25* and u(t) is the velocity of the ball at time The gravitationa constant is g 9.8mls?a) Find differential equation for the velocity v:(-0.25/4)v-9.8b) Solve the differential equation in pan a) and find f

point) body of mass kg is projected vertically upward with an initial velocity 31 meters per second. We assume that the forces acting on the body are the force of gravity and retarding force of air resistance with direction opposite to the direction of motion and with magnitude clv(t)| where =0.25* and u(t) is the velocity of the ball at time The gravitationa constant is g 9.8mls? a) Find differential equation for the velocity v: (-0.25/4)v-9.8 b) Solve the differential equation in pan a) and find formula for the velocity at any time u(t) 187.8e^(-t/16)-156.8 Find formula for the position function at any time if the initial position is s(0) s(t) = 3002.88(e^(t/16)-1)-156.8t How does this compare with the solution to the equation for velocity when there is no air resistance? If c = 0 then v(t) = 31 - 9.8t, and s(0) = 0, then s(t) = 31t - 4.91?_ We then have that U(t) = 0 when ~3.163 and 5(3.163) ~ 49.031, and that the positive solution to s(t) = 0 is t ~ 6.326, which leads to 0(6.326) 31 meters per second



Answers

Involve vertical motion and the effect of gravity on an object. Because of gravity, an object that is projected upward will cventually reach a maximum height and then fall to the ground. The equation that determines the height $h$ of a projectile $t$ seconds after it is shot upward is given by $$ h=\frac{1}{2} a t^{2}+v_{0} t+h_{0} $$ where $a$ is the acceleration due to gravity, $h_{0}$ is the initial height of the object at time $t=0,$ and $v_{0}$ is the initial velocity of the object at time $t=0 .$ Note that a projectile follows the path of a parabola opening down, so $a<0$ An object is thrown upward and the following table depicts the height of the ball $t$ seconds after the projectile is released. Find the initial height, initial velocity, and acceleration due to gravity. $$\begin{array}{|c|c|} \hline \text { t Seconos } & \text { HeiGHT (FEET) } \\ \hline 1 & 36 \\ \hline 2 & 40 \\ \hline 3 & 12 \\ \hline \end{array}$$

So the Cuban christian is a hopeless new york city plus a by two T spare because to etch. And the table presented is as follows tea time and its height in feet. The time is one country and high forgiveness 84-136-10 1 56. So just write down the equation at respective time because to one A chocolate no one place it's too awful and experience what Because to 84 equals two actual plus Neil two plus a way to despair is born. It was 2136 And he was three. A topless video three less able to Of nine is 1 56. They know this as a question one equation to an equation three respectively. You know Question of 2 -3, you're one plus subjecting this side. He went to three equals 2 52 and question 3- stick person too. This can be lieutenants you want plus able to 4 -59 minutes for this fight. So this is 21 56 -136. Let this big question for And this big question. So the question 5- situation but now will be a right to you will be canceled on As it is same ever to back at two years 20-52 For a finally themselves to 32 feet force again square. Mhm. Since we have known A. So by the question of four you can see he was two into three In 23 plus This video quantity was 52. So we a little bit particular minus A into 3 2. And a value we have calculated just about Here is -32 50 to -32. Into the way to Which is 50 to plus 48. Or we can say 100 1 300 ft or second. So the US 100 ft per cent. And lastly that term Actually. So from the question one we can say A topless real one plus a white two of one was 84. We have known the two values. We always 50 100 and he is minus attitude. Yeah. and one is 84 I took plus this quantities 100 16 was 34. This is a joke. Last 54. Close to 84. And these councils. How So experience? Finally, only 05. So he had out of the answers, you know, Minor strategy to quit for a second spirit. Mhm. We not 100 ft was in And it's not this cedar city. So this is our

All right, So this question deals with vertical motion with gravity and acceleration due to gravity. So this is the equation that represents that age of T equals 1/2 a T squared plus B T plus age. They give you a data chart with some data collections. And so the first relationship is 1 84 which means that at Tom one, this projectile is 84 feet near. I also give you two and 1 36 which tells me that at time to its 136 feet in the air. And then finally they give you three and 1 56 that a time. Three. It's 156 feet in the air. Your ex coordinates in these ordered pairs are your T values, and your y coordinate is your h of tea. So we're gonna substitute those in to create our system of three equations. So if I start with this first point, I'm going to have ah, 1/2 a time to one square. Excuse me. I'm going to drop the zeros and just say v times one forwards sake of time and then H now if I clean that up that's gonna equal 1/2 of a plus B plus age. And then it equals the Y value, which is 84 for this one. If I put this one in this time, I have 1/2 a times to square was V Times two was age, which is 1/2 of four, which would be to a plus two b plus h equals the Y value of 1 36 And then finally, this point here is gonna be 1/2 a times three square was being times three plus age, which will give me half of nine or not haves a plus three b plus age equals 156. And we've created our system of three equations, three variables and three equations. And so now we need to start the process to solve that system. So I'm going to start with the first and second equations, so I have 1/2 a plus. The close H equals 84 paired with to a plus. To be plus age equals 1 36 I could immediately subtract those because my h is air the same. And so that gives me negative three have a monos v equals negative. 52. All right, We'll do that again, this time with the first and the last. So I have 1/2 a plus. V plus age equals 84. Repairing that one with the non hands A plus three v plus age equals 1 56 Since the h is air, the same on this case is well, can immediately subtract. This is gonna give me negative for a monness to be equals Negative 72. This has reduced my system now into a two variable system. So we're gonna put those two together. So we have negative for a I'm honest to be equals negative 72. If I multiplied this equation by two before I bring it down That way my fees will match up. So I'm gonna have negative three a modest to be equals negative 104. And so then subtracting is going to give me negative a equal to 32 and so a equals negative. 32. Once you get one of the variables plugged that into one of those to bury league rations, we have negative for a minus to B equals negative 72. So negative forward times. Negative 32. Multiply that and then subtracted over. You get negative to V equals negative 200 and so v equals 100. Now that I know A and B use one of your original equations I'm gonna use to a plus two V plus age equals 1 36 and plug in your A intervene so two times negative 32 plus two times 100 plus h equals 1 36 Multiply and add Move it to the you're gonna get 1 36 plus h equals 1 36 And so, if you subtract h zero So our equation then, um are what they asked us for. Here the acceleration is negative. 32 The velocity and Tom zero is 100 and the height at Tom zero is zero from there are three unknown quantities.

All right, so in this question, we're talking about vertical motion. So this is the equation for vertical motion, and then they give you a chart with some data, and so we want to pull off the three points that it gives us. One of those points is the point then of one and 36. And then there's also the 0.2 and 40 and the 0.3 and 12. Now, each of those points represents an X or I should say, a t at an age of tea. So we're gonna plug in the X or the T values and then let those equal the Y values or the age of teeth. So in this first case, we know this is 1/2 I times one square plus V zero times one plus h zero, and it should equal 36. So when we clean that up, one squared, of course, is just one. So this is one hand I plus v sub zero, which is the velocity of Tom zero plus h subzero height at time zero. And I should equal 36 okay, for the second relationship, the exes too. So we have 1/2 a to square plus V sub zero times two plus h subzero. And this time it was 40. So cleaning that up, we know that two squared is four times 1/2. Makes that to a plus two times Visa zero plus h subzero equals 40. And then finally or third data point will allow us to get 1/3 equation. And so this time we're gonna put the three in. So we have 1/2 a three square plus Visa zero Tom's three Bless age. It was 12 nine, divided by two. This is gonna be nine haves A plus three. These of zero plus h zero equals 12. We now have three equations that we can now put together in a system and solved. I'm gonna take the first to the red and the blue and pair them up and eliminate my age is because you noticed that the H is the same in all of them. Now I'm gonna drop the sub zeros just to make this easier to write. But I'm gonna take 1/2 1/2 of a plus. The plus h equals 36 and I'm gonna pair it with to a plus to be plus age equals 40. Since the ages air the same, we can subtract. And if we do that, we get 1/2 minus two, which is negative three halves. A modesty equals negative for going to repeat that this time with the first and the last Someone to take 1/2 a plus. The plus age equals 36. And I'm gonna pair that with not half stay plus three V plus age equals 12. The ages are still the same. So I'm going to subtract him when I do that 1/2 minus nine halves is negative or okay, one minus three. So negative to the equals 24. Now I've reduced it to a system of two equations. So we're gonna put those two together. This one says, negative for a I'm honest to V equals 24. So if I take this one and multiply it by two, I'll get negative three. A modest to be equals negative Eight. Since the two these are the same. We can subtract. Negative four plus three is negative. One and 24 plus eight is 32. So a equals negative. 32. Now that I know a I could go back in and around three. And so we know that we had negative three halves. Hey, Mona's V equals negative for so negative. Three. Hap's Tom's negative 32 Monness v equals negative four. Which means that negative B equals negative 52 and V equals positive 52 with multiply and then subtracted over. I've got a of that be I need h So take one of your equations to a plus. To be plus age equals 40 two times negative 32 plus two times. And this should be negative. No, it's positive. 52 Sorry plus H equals 40 multiplying adding you get 40 plus h equals boarding. And so age equals zero. And so in this problem you were asked to find the acceleration. We know that a is negative. 32. We know that Visa zero is 52 and the ages of zero is zero

Mhm. Okay. What we're gonna do is we're going to walk through a kind of like a word problem. Um We're starting with an object that is projected upward from ground with an initial velocity of 500 ft per second. So this is a vertical motion problem. And the first thing we want to do is we're actually going to be neglecting air resistance. And what we want to do is to determine the velocity equation. And so we know that um the velocity equation generically is our velocity is equal to our initial velocity plus the acceleration times time. Okay, so for our purposes um The initial velocity we're told was 500 ft/s. And of course acceleration because it's vertical motion acceleration is due to gravity, so that is negative 32 ft per second squared. Um So it's going to be negative 32 times T Okay, so there is our velocity function with respect to time and now what we're gonna do is we're gonna actually graph um this velocity function and of course um this velocity function is linear. So this is an equation for a line. But we're gonna go ahead and grab him using a graphing calculator. So I'm gonna switch over to my graphing calculator. Dez knows um is it online, free online graphing calculator. Um And we notice already have it in Dismas. Um And so I'm actually going to um Zoom out. Um and so you notice of course the Y intercept is right up here at 500 and it's a decrease in line. So this is what that decreasing line looks like right here. And now I'm gonna go ahead and quickly sketch my graph. And so if this is T. And this is my velocity Moded at 500. And then somewhere here around 15, a little over 15 is where it crosses. So we're just gonna go ahead and do that. Um Okay, so that is the velocity function. And so now the next thing we want to do is we want to go ahead and use our velocity function to find the position function. And so remember, the position function is equal to the integral of that velocity function. And we're integrating of course, with respect to time. Okay, um and so now we're just going to integrate this. Um and so my position function Is equal to 500 t -30. Open note is not going to be 32, it's gonna be 16 T squared plus some constant of integration. Okay, so now we're going to find that constant of integration. Right? And so what we know is The object is projected upward from ground. So at time zero at a time Zero, my position is on the ground which is at the zero level. And so that is going to be equal to 500 Time 0 -16 times zero Squared. So that is telling me that constant is zero. So really my position function is equal to and I dont know why thats freezing up on me is 500 t minus 16 T squared. And there is my position function. And so now we want to know when at what time do we reach maximum height and what is our maximum height? And so remember, time at max height is when my velocity equals zero. So we're going to set our velocity equal to zero and we're going to sell for that time. So that tells me t is going to be equal to about 15 .625. And so now I want to know what my maximum height is. So I am, my max height is equal to my position function Evaluated at that time of 15.6- five. And of course this is in seconds. And so this will be 500 times 15 0.6 Too far to five. I don't know why that's doing yet -16 Times 15.6-5 squared. And so that will actually be equal. If you put that in your calculator, that is actually going to be equal to 3000 906 point to five ft. So there is my max height. Okay. And now what we want to do is um now we're gonna look and this is if I neglect air resistance. Right? So now what we want to do is um if we're told if air resistance is proportional rips to the square of velocity, okay? Um and it's going to be given by this equation that the derivative of the velocity is equal to negative 32 times. And that should be on the outside. So negative. And in parentheses we have 32 plus K B squared. Okay. Um and we want to use, we want to actually solve this integral equation. So we have one over 32 plus K V squared. D V is equal to negative the integral of D. T. Okay. Um this denominator almost almost looks like an arc tangent. That the problem is this K. That is attached to the U. Squared. Or in our case R V squared. So if I factor out a K. So I'm a factor out. I want over care. I get the integral of one over 32 over K plus B squared. Now that definitely looks definitely looks like an arc tangent now. Um So this integral of DT Okay. Um and so that definitely now looks like one over a squared plus U squared, which is an arc tanne. Right? So um that integral um is a one over Okay. Times and of course the integral um of one over a squared plus U squared is um one over a times arc tangent of you over A. And so that will actually equal negative T. Plus. See okay, so now what we want to do is kind of rewrite this because we want to solve for velocity, we want to have this in terms of velocity. So we're gonna kind of clean this up a little bit um and get velocity by itself. And so we and I do that. Um We end up with this actually becomes one over The Square Root of 32. Okay, and then this will be arc, that's two arc tangent of and then this is the square root of K. Over 32 times fee. Um Equal to negative T. Plus E. Okay. Um and so now we're gonna multiply both sides by that square with just 32. And we're going to solve for T. I'm 40, I'm sorry. And so when we do that we get the is equal to the square root of 32 over K. Times tangent. Because remember we've got to do the inverse of art tangent and C minus. And then we have the square root of 32. Okay. Times T. Okay. Um And now we need to find out what C. Is right. We want to find out what C. Is. And so we want to go ahead. And what we do know is when t when T equals zero, then the Is equal to 500. So if I use that concept and save 500 is equal to. And then We put in here the 32 over K times tangent. So that's just going to be equal to tangent of. See because T. is zero and so we get C. Is actually going to be equal to Yeah. Arc arc tangent. I don't know why this is messing up. Arc tangent of 500 times the square root of K. Over 32. Okay, so now if I put that all back in here we have the velocity as a function of time is equal to Yeah, Is equal to. And then we have this big old long thing, we've got the square root of 32 over K. Times tangent of C. Now is arc. Mhm. Are tangent of 500. Okay. over 32 and then that is going to be minus The Square Root of 32. Okay, times there we have it. That is a big old long crazy mess. Um And then what we want to do is we actually want to um use a graphing calculator. And we're actually gonna graph, we're actually gonna graph mhm. V. F. T. When K. is equal to .001. So we're actually gonna put K. In there as 0.001. And we're actually going to graph VF. T. And from that we actually want to approximate the time zero in which The height this maximum. Okay so wherever there's AK. We're gonna insert .001 and we're going to graph it. So I'm gonna switch back to um my online graphing utility. I'm gonna turn off that linear function. And you notice I already have um this function here right here. And I put in um Where there was, okay, I put in .001. And so I end up getting um I end up getting let's see um 32,000 square to 32,000 times tangent of art can um of 500 times 5000.3125 minus that 0.32 times the X. Value for the T. Value. And here is here is that graph and I'm going to be focused on this section right here. And so you notice that where my height is going to be maximum is the area under the curve right? Um And it's going to have to be a positive area under the curve. And so if I just kind of look at this graph, if I keep blowing up this is kind of like a tangent function. So I know where that area is going to be. Maximum is in this little triangle spot right here from zero to some X. Value. Or in this case t. So that is telling me that maximum height is going to occur when t. Is equal to 6.8 six. So if I switch back over here whips where'd it go? So if I switch back over to my right board. Okay. So um so when K equals .001 um max max height will occur at T equal to 6.86 seconds. Okay. Um and now what we want to do um is used an integration capability to actually find that max height. So we're actually going to integrate from 0 to 6.86 of this function right here. Okay. Um and of course um with K Equal 2.001. And so we're going to actually integrate and we're not gonna do it by hand. Of course that would be totally crazy. So we're gonna do um integrate of I think it was mm 32,000 then it was tangent of are tangent Of and I think it was 500 times the square root of point 1234 and I think it was 3125. And this was all in the arc tangent minus the square root of .03 two T. T. T. Okay, so we're actually going to use the graphing the capability um integral capability to integrate. Um I already have this in here so I'm actually I'm going to I'm just actually changed this and I'm going to do in control from zero. That's the neat thing about gizmos 8 6. And then of course I've got to add in my um the X. Here. Okay and that is saying my max height is 1088, so 1,088 ft. So based off of the integral, so this is about 10 88 feet. And so why is there such a big difference between what we found in part a versus what we did here? And the big difference is in part a or the previous owner where we neglected air resistance. Okay, so in in the previous part, we had no air resistance. And remember, air resistance is going to impede the vertical motion of your object. And so since this latter part, we actually inserted air resistance, we would not expect the object to travel as high because of the resistance of air resistance, the resistance of the air friction.


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