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Abii 25 Q1) At a football stadium a goalkeeper kicks a football at initial velocity of 16 mls in the vertical direction and 12 mls in horizontal direction (a) At wh...

Question

Abii 25 Q1) At a football stadium a goalkeeper kicks a football at initial velocity of 16 mls in the vertical direction and 12 mls in horizontal direction (a) At what speed does the ball hits the ground? (b) What is the maximum height that ball can reach? (c) How long does the ball ?remain in the airJill

abii 25 Q1) At a football stadium a goalkeeper kicks a football at initial velocity of 16 mls in the vertical direction and 12 mls in horizontal direction (a) At what speed does the ball hits the ground? (b) What is the maximum height that ball can reach? (c) How long does the ball ?remain in the air Jill



Answers

A ball is kicked with an initial velocity of 16 m/s in the horizontal direction and 12 m/s in the vertical direction. (a) At what speed does the ball hit the ground? (b) For how long does the ball remain in the air?(c)What maximum height is attained by the ball?

For this example will be doing a projectile problem where we kick a ball with a horizontal velocity of 16 meters per second and a vertical velocity of 12 meters per second. So the net result is that we get some trajectory that looks like what we have above here. What goes up like this travels in an arc and then comes back down like this. I don't get so what we want to figure out is what is our final velocity? How much time does it take for the ball to travel this path and once the maximum knighted Regent's So we want the final over here, right? The magnitude of that final velocity vector this distance here it will call age, and we also want how much time it takes tow. Um, go along this path to don t OK, so for the first part, the final velocity, what I would do is recognize that this problem is very symmetric, meaning that we get kicked up. We should speed up by a certain amount, reached our maximum height and then we're going to decelerate, uh, by the same amount in the opposite direction. So we should go up with a speed of the initial for going, you know, end up with a speed of zero up here, at least vertically. Right? So I should really say that V at the R V y at the maximum height is equal to zero, and then we start accelerating again in the Y direction and end up with their final lost me here and the thing. The thing to realize is, since this is symmetric situation, that the initial Y velocity is going to be equal and opposite to the final lie velocity. And we already know that the X velocity is never going to change because gravity only acts downward. Right, So this is the same all throughout. So really, our final velocity doctor has a why component. Ah, minus the initial y velocity and the same explosive. He is our initial letter. So the magnitude of this factor is going to be the same as the magnitude does of our initial speed, Becker, Because we'll be squaring both of the components, so any sign differences will go away. So we not would be, you know, why squared plus B not X squared. And that's gonna be the same as the final velocity because we're just taking minus. We know why a squared plus B not X squared. So if we go ahead and calculate with the magnitude of the not is just by plugging in our values there, get that it's 20 meters per second. And we just argued that this is equal to the final velocity as well. Now, if we didn't want to make our argument this way, we could just figure out what our final component in the Y direction is going to be from our cinematic equations. Specifically, if you recall one of our equations, it's that the initial velocity and the wider actions squared equal to the sorry that should be the final to the final velocity in the wider action squared is equal to the initial velocity in the Y Direction square minus two g times. The difference between why and why not? So the def. So why and why not? Are both going to be zero for us, right, because we start at zero, which were defining as the ground and we end up the ground. So this whole term goes away and we would see that our final velocity in the wider action. It's just gonna be equal to our final velocity squared will be equal to the initial velocity squared and the only solutions to this or that be not why is equal to plus or minus the final in the Y direction. And we know that it's gonna be negative because it's going downward. So that would have been another way we shouldn't that out. But just a good thing to know. So now let's look at the total time that it takes to travel this trajectory. But this we're just gonna use another one of our kingdom attic equations that states. Why minus why not people to you know why Times t plus 1/2 a T square we're in this case will be using G. Jensen, and it's just gonna be the acceleration due to gravity. Okay, so now, similar to above, we're going to be saying that Why not? Where we start and why? Where we end up are both zero since we're trying to figure out the time it takes to get to the ground again. So what happens is we're left with this equation. Do you know why Times t plus 1/2 gt square where we can factor out time from both of these terms. We're at least one T, and what we'll have is this quadratic here that has multiple solutions. So one of the solutions is, of course, going to be t equals zero. And that makes sense because you could imagine if we let why just be equal to our starting position? Yeah, of course. It's not gonna take any time at all for us to just get our starting position because we're already there. So the other solution, which is gonna be the one we care about. That's what happens when this factor goes to zero. So specifically, we're looking at you know why? Plus one have t t equals zero. So we're gonna subtract vino y from both sides. We'll have 1/2 g t equals minus view, not lie. We're gonna multiply both sides by a two over ji. So we should have two times Vietnam. Liar, Reggie. All right, so we have that. And sorry. We should have a minus on this other staff here. So we have, you know, wire, right? We know it's just gonna be equal and opposite to our original. Um, sorry, No, we were just given being outline. And we know the G is minus 9.81 meters per second squared, so we can just go ahead and calculate this time. If we do that, we should get 2.45 seconds. All right? Now, for the last stretch here, we're gonna figure out our maximum height. Now, this could be determined a couple ways, but there's a common the equation that we can already just utilize for the maximum light instead of Soling for it ourselves, which is that the maximum height is just gonna be given by our initial velocity in the Y Direction square divided by two times the magnitude of gravity. So we already have our initial velocity in the Y direction and we have gravity. So we just have to pluck these values that if we do that, we'll end up with H equals 7.34 meters. So course the way to solve this manually without knowing this equation would have been to look at the situation in which our ball reaches the maximum line, which is characterized by our velocity in the Y direction being equal to zero. So we would be solving for the height in the situation in which our arrival Aussies equal to zero and we would have arrived at the exact same equation that we have here. So there you have it. Um, in this projectile problem, we determined that our final velocity was going to be the same as our initial velocity in magnitude. At least we determined what how long it would take for our ball to go up in the air and come back and hit the ground, and also with the maximum height that our ball reaches is

So we have a soccer ball being kicked. And that's the formula for the path of the soccer ball. Age of us represents the height X represents the horizontal distance. And our job is to find the maximum height and the horrid horizontal distance. Now, while these numbers are not easy numbers to use, I plotted it on my graphing calculator. So what we need to know is the maximum height. So we are going to hit menu and we are going to find the maximum point and that point is somewhere between here and here. So the maximum height is 39.06 ft. Because that is oops, I lied. That's not a maximum height. That should be my horizontal distance. Now, the maximum height Is the Y Value on Here, And the horizontal distance is how far does the ball travel? So we don't want the horizontal distance right there. We want the horizontal distance that the ball actually travels. So I need to know from here to here. So I am going to look up the Um analyst graph. I wanna look up to zero function or the zero point. And if I do that, I get 78.125 ft. And that is my horizontal distance.

Hi students. Today we are looking at quadratic equations and modeling them in real life situations. So we're looking at the height of a ball that it's punted and is given by this quadratic formula. First, let's talk a little bit about the form of the formula is incentive for which, if we look over here to our notes, we see that that means that the highest degree is leading with a two. So the first term is X squared. The coefficient is a of X square. The coefficient of X is B and the constant SC. So that's gonna be there important as we go throughout today. Also, we're gonna use the good old quadratic formula to solve our problem. Okay, The first part, How high is the ball when it is punted? So when you first put the ball, imagine you're starting at a start point. Let's say that you're starting right here, Okay? You're punting the ball as you punt, the ball of the borrow goes up and then it comes down, OK, so it will continue on the other side possible. Let's just worry about this portion. So when I start the point, I'm starting at a number of X equals zero. Okay, So when x zero, that means that these two terms will cancel out and you'll be left with your coefficient. Here's a little trick when you're dealing with the quadratic formula and it's in standard form. The constant is the Why interested. So if you could go back to your memories about linear functions, y equals MX plus B, the constant was be and in this case is very some of the constant SC. But let's say you don't trust that and you want to do the math. It's not a problem. If you do the math, you could just write your function well, right. It will click right here and you can put plug in zero for X. Because remember, you're starting at a point of X equals zero. You're starting here? Sure. Why interested? Okay, then, look. Look what happens, bam, That cancels out and you're left with 1.5. So your first answer is 1.5 feet. That is your Why interested. Okay. And that is also your constant. So that's the answer from letter it. We're doing great. There. Looks like a letter B. What is the maximum height. So let's look at our good old quadratic equation. This is called a probable when it's on a graph. The maximum height is that Vertex? So your text and remember you're labeled is H and K. Okay, which is your it's now you can you rob out. So you want to find the maximum height that that's okay that on the morning to do that, let's go back. It's our memory banks. And actually, if you have a partner and formally you could see a portion of the height from your quadratic formula, what we're gonna use is over, and we know from our let me redo that again for them to redo that it was a little bit bad. There we go. So we know from our previous equation we know what a, B, A and B are A is going to be negative 16 over 2025 the is going to be 9/5. Okay, so I went ahead and pulled up dismal, so we can already kind of have it in there. So if I plug this into the formula, I get this handy dandy equation. I love decimals because you can put a lot into the equation, and you can see it is the decimal form or is a fraction for. So let's leave. It is a fraction form. So that answer is going to give me 3645 over 32. Okay, once I get that answer will come back to the white board. We get 3645 over 32 then I'm gonna plug that back into the formula into our original equations are gonna plug that into our equation. So once we plug back into our equation, here's another decimals already plugged in. As you can see, this is the same equation as what we see up here. I just plugged it right back in. When I plug that back in, I get 6657 over 64. Okay? And that is gonna get K about. So if I come back and plug it in, I get 6000 inches. Letter B 650. Let's check again. I think that was 54 57 over 64. Okay, that is going to give us is a recap. The maximum height. So this is our k value. So again, when you're dealing, look are dragged its and you want your Vertex you find hnk in order to find h you do minus B over to a That's gonna give you hh in order to find K. You plug that back in. So this is your input in case you're output. So we're doing great guys, we have a only half be last question. How long is the punch? Uh, what do they want? How long is the punt so typical? You have a question like this, you want to get the solution of the plant. So I see I started way over here and I ended up here. Now if I were to go beyond, let's just do a little hypothetically, I was to go beyond. I will see that my parable a has a negative value any positive about, Okay, But we really want to get the positive value because we're looking at length and distance. Distance is always positive, is never negative. You don't hear about anybody kicking it negative. 50 yards. Okay, You hear about the puncher kicking it 50 yards. So in order to do that, come on. But so are handed any quadratic formula. Now again, we have done great Sonora A, B and C. And that's here so we can plug a B and C back into our quadratic formula. So we have our car genetic formally here again, Dez Mo's wonderful, wonderful tool. Because when I have gizmos bam, I can plug everything back into decimals and get an answer done for me. You just have to be careful of your negatives. Now you may be asking Wait a minute. What happens to the plus or minus? Glad you asked. Well, here I want trouble. So in this case, I tried the plus. The plus gave me a negative point. Remember? We talked about that. Distance cannot be negative, so I don't want to do that was so let's go with a the negative down, okay? Or Leggett The minus time. Oh, here we go. We haven't ESA Our answer is 228.64 I'm running enough to the 100. Okay, so when I come back in, I have my answer as 228 0.64 And again this is crimes of the hundreds. If you wanted to run to the nearest whole foot, it would obviously be 229. And that would be your final answer. So again is a recap. The first thing we did is that we look for the heights, the height when the ball was constant, which was our Why interested? Okay, so that was our Why interested? Alright, right. Interested here. And we did that just by looking at the constant there for the maximum height the Vertex We used the formula of the over minus B over to a We did that. We got our answer and plugged it back into our equation to get R K value. Then for our final problem, find a portion of our problems. Said how long was a plan? And that is the solution. So this is a model that you can typically follow for these types of questions. The quadratic equations, especially project Our motions are very predictable. And you can follow this for your problems. Okay, guys, Thanks so much. Not talk to you soon. Bye.

In fact, this problem. Yeah. The printed football across the situation behind five and 16 over two in 2025 and x squared. What exists? Good. Horizontal distance. 9/5 X. This one. Right. All right. So when the football hasn't traveled at all, right, Because I guess there's the football. It starts off the ground because, um, the no, he is just planting the ball. So it starts off the ground. He just dropping in or whatever kicks it. Why is a certain distance and then lands on the ground, right? All right, so this distance right here is when x is equal to zero, right? Thanks. Zero just access equals some big number. We don't know what it is yet anyways, so one X is equal to zero. This becomes zero this time, become zero. And when we get, we're left with y is equal to 1.5. So it is kind of the initial head of the football for Part IV. We want to find the maximum kind of the pump. So to do that, we do negative, be over two times. They, um All right, So that's B is ninth over five So we have over five time to 16 over 2025 right? So that that's a native 16, particularly 25 that is equal to known. 25 founded by, um five times two times 16. Oops To put 182 to 5 over. Um, uh, if I can students tens, that's 160. We brought Simplify this a little more 3645 invited by 32. And I think that is about as simplified as it will get. Lead it at the for now. All right, what is the horizontal distance that the ball will go? So, really, that means we want to find when or why equals you're right week with zero At some point. I guess that doesn't really make sense, since he he drops the ball. But if the ball had continued from the ground up through where his foot's gonna kick it, go through this year right here. It's also gonna go through some point down on the field when y is equal to zero. So if we go back to the original equations and put in zero for y, we get zero secret too native 16 over 2025. Word. It's a member of faith. Thanks. That's when we drive. That is a quadratic equation so we can use the quadratic formula. Yeah, negative. Be right here. Plus or minus the square root of B squared. So B squared is this square, which is 81/25 uh, minus four times a, which is another negative numbers. One negative times a negative is positive. So I just put it in there now, Uh, time. See 1.5 already then. So that's get putting into that toward calculator here. Um, so we'll take a brief moment you can either along. Put it in your own calculator. Uh, take it quick. Now. Whatever floats your boat, I guess. Alright, We've almost got a got an answer here. Long as I put it in correctly. You never know. All right, so we actually have two inches here, right? Because it was the plus or minus right there. So the first one is negative. 2.8303 Here's the first one. Second one. Whoops. Second one is 228.64 All right, which one of these is the answer you want toe put down, Uh, your homework? Well, it depends on if you want to get the right answer. And I guess, but a better answer than that. Let's go back to this picture right here. Remember? Always said, this is X is equal to zero. This is like native somebody get a number. And this is some huge positive number, right? Look at that. Is that a coincidence? Absolutely not. Eso This number actually is kind of irrelevant to the situation. It would be as if the ball had chart from the ground. Um, which it didn't. It started out the kickers foot, right. It's already kind of part way through this problem. This is the real number that we want. 228.


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