So in this problem, we're going to use some Kine Matic equations to figure out the figure out the Parametric equations of of e b ball that is kicked with respect your time and so and so we have this. This position of X is equal to the initial velocity times the coast I have the angle of times t plus the original the original starting position X and then so for y t. Since since our Y component has grabbed the acting on it So we have this negative 1/2 a cheese square buzz be not sign A t, which is the initial velocity plan is a sine of the angle attempts t plus the initial initial Why by position and so so if we use this well, we can say all right, What we know the you know that the X component should be X is equal to X is equal to We have this initial velocity of 1 40 times. He co sign angle of 45 45 times t and and this problem we can assume that that our initial exposition is euro. We have this X is 1 40 coastline 45 times tea. Then we have. Why is equal to Well, we have this negative 1/2 of the of acceleration, right? So in in acceleration in terms of feet is negative is 32 sides negative 1/2 times 32 which is negative. 16 the negative 16 t squared. She squared plus again, you in your velocity 1 40 This time we have sign of the angles or sign 45 times two plus our initial life commission, That is. He danced in the u zero. It was. Look at this graph. We look at this graph. Well, we can figure out when the exposition is. We can figure out what the exporting it is right here. Yeah, if we figure out where. Well, this is why component is zero, and we can figure out where the Y component is. Zero, If we re say Okay, well, let's factor out 80 the same tea we have. Negative. 16 plus 1 40 sign. 45. This people, this is the negative. 16 t negative. 16 to you. Plus 1 47 45 This should equal zero. And so Well, we already know what we have. These two points T is equal to zero. Then we have to solve for the inside negative 16 TV is so making 16 t plus 1 40 signed by is equal to zero. When we do that, we do 1 40 45 divided by 60 we get six point. He is equal to 6.2. You see what you 6.2? And if we plug in 6.2 x what we get that our exposition after the ball hits the ground. So remember, the ball goes, uh, and then it's the ground. So when X is at this yes, 6.211 time is six point to This should give us her our range. So how far how far did this ball go? This is This is what we're trying to figure out right now. How far do this ball coach? So let's play and much. 6.2 13 to find how far the ball went. So get 6.2 times. One for you. Co signed 45. So this X is 613 point white age eat. Let's late with this. This is seconds. This is also seconds. And so we found that so So if a T is equal to six point to the ball hits the ground again. Well, if this is a parabola, then we know that at some point along here, half of the distance half of the time because it takes half the time to go all the way up to its maximum height and then half the time to go back down. So well, if we divide 6.2 by two, get 3.1. So what? We know we hit our maximum height. AT T is equal to 3.1 seconds. And so if we saw a full if we saw for why, what we get is negative. 16.1. Thus, 1 40 five 3.1. What this gives us is 153. So So our Max Max height max height is 153. Why one It's this is distance, family distance travel. And so so this. Yes. Yeah, is Max I travel. And so let's see. Let's see if we're missing anything. Well, we found we found Max height. That's a check. We found the distance traveled. We even found we even found our parametric equations. right here. So we have this permission equation Eggs of X with respect to T and I will respect t so we have effectively solved our problem. We have solved department, remember? Remember the formulas we used this these are These are dramatic formulas for position of a of A ball and it uses, uses gravity, uses initial velocity and use this These two equations are very, very important problem. So I've labeled labeled what each each very mo means. Remember, this is this is the trajectory of the ball. And we use reasoning to figure out that that if we can solve for for our endpoint here when the ball hits the ground in other words, were y zero where are wife of one of the ball is your way solved for that? And our X component is when is at the time where the ball hits the ground. So this is our distance travel. I'm liking in 6 22 unplugged in 3.1 because it's half of 61 too. So somewhere between those have half the time it takes for the ball to go up and back down. So we want to figure out where those maximum up and this maximum up our maximum wise A maximum height which we figured out to be 1 53 So we've checked all of our all of our questions and we are done.