All right, We have reached good old problem, number 70 and this is probably gonna be your favorite problem with the whole chapter. Released his mind because we finally get to see some real world application of all these pyramids questions that we've been solving. So here we're giving a a good little physics problem. And it's we're told that path of a projectile is given by these two questions. Got X is equal to the subzero or vino times coast on if they all times t and why's he could h plus B not Santa Times T minus 16 square And here X is gonna be the horizontal distance from where the projectile is launched Toe where it is now And the no is gonna be the initial velocity that the projectile is launched It data is the angle that the projectile was launched T is gonna be time in seconds. Why? I was going to be the height of the projectile at a given time. H is going to be the initial height or the height above the ground that the projectile is launched and recorded. Talk about being on They dont So we're told in this case we have a projector, the baseball hits gets hit by a bat, has given initial velocity or V not of 150 feet per second, and it's launched at an angle of 35 degrees from the horizontal. And it was hit an initial hot three feet. So let's draw little pictures here. So here's our ground. And here's our a little man with a baseball bat and he hits a baseball and it's gonna go flying through the air. So he did it right here in a hot three feet. Lost it like this with a velocity of 150 feet per second initially, and it was an angle fada of 35 degrees from the whores only. This is a situation that we're doing Bill with and part of a ask us to find the Parametric equations that describe the position of the ball as a function of time. You really only have two. Here's Blufgan. These babies, Avi, not they didn't h. And to get are very much in question. So I have X is equal to the note, which is 1 50 come to coast on of 35 which is all times t Now, if you think about this little sickness report here is always gonna be a constant. This 1 50 coast on 35 Mr Costin, and lets that calculator quick like clearing the right mood. That's about 1 22 point. Just keep him. I just got this cup Sitting here is not a variable. It's all just a constant Cumpsty. And now he's gonna equal H, which is three plus one. They sign of 35 degrees times t my 16 hey, squared. So there's our set of parametric equations that describe possession of the ball is a function of time. And I think about this, which Collins has given us his forgiven time t go back to her picture here. Just gonna give us the X and Y coordinates. If we assume that this is the Y Axis, this is the X axis, but the origin being right to the point on the ground directly below where the volume was first hit and Part B, we wanted to describe the ball's position after 12 and three seconds and locate Those solutions are curved, so basically we won't know for T equals one second two seconds three seconds. Where is the ball? What's his coordinates? X and y. So we've got no, I do. Here's plug in the values of team and get out. Um X and Y coordinates next equals 1 22.9 times T times one just 1 20 Do that. Times two for tea is to 45.7 and then times three iss 3 68.6 How for while we've got three waas 1 50 Signed. 35 terms T rules one. Not now. Vice 16 once. Where in that? 73. What? Oh, around to the nearest. And then we'll change that to take one too. Again t to their get. Why 111 one And finally for t equals three seconds. Yeah. 1 17 What? One. And now it's graph that Yes, I get the gas going bear y axis and our X axis. So we've got Let's call this 50 could be 100 1 50 200 to 50 300 3 50 400. Thanks. And er Oh, this 25 50 75 100 1 25 1 50 one's head 200. So let's just bought this point. You've got 1 22.9 Here. 73. Yeah. 2 45.7 Get 1 11 should be right there. And 3 68 the year. 1 17 for one, which is not much higher. So we're on right there. And so this is the path for Bo gonna be a problem. Which now what we expected, I guess. Really. This shooting girl with zero? This is goingto Michael's three feet when X equals zero. All right, so there's part beef part c asked. How long is the ball in flight? So how do we figure that out? So we won't know this way. So this Paul is gonna come back down, gonna hit the ground somewhere here. I don't know how long that takes. So if wow, when the history Graham's gonna be zero. So really, we need to solve this equation here and figure out When? What Time T that while legal Zero. So three. So we're going to hear zero y zero three plus 1 50 It's on 35. Freeze, Humpty. My 16 t squared. Let's rewrite that just to be a little. We're clear. It's perfect. Put this in the quadratic equation to solve this. All right, So what? Values of tea? Obviously, one of them is gonna be close to zero, but not quite zero. Because we know Tom zero. Why was actually three. So let's see, we got the quadratic equation, right? Say to use it to negative B. This is our B term native 1 50 signed. 35 degrees plus or minus the square root of B squared 1 50 Sign 35 Reese Hall Square minus for Time's a his native 16 Tom. See, which is three over to a two and a was negative 16. So I'm gonna pose a video here and take just a second to run this big thing through a handy dandy calculator. All right, coming back here. We're seeing it will be solved. This for the positive solution we get. T is equal. Negative 0.3 seconds. Obviously, this ball is in the air a little longer. Point over three seconds. What this is is saying that and I can't have native times over. This is just extremist altogether. But what this is saying is that when Tom Mickle zero, you're three feet. So it just before that, if you were a girl extend this curve out that you get on the ground just a little bit prior to zero. Is all that saying That's just extreme solution. So, really, the ball's in the air for 5.4 seconds. And the second part of part C asked, What's the total horizontal distances? Terrible's what we know that X, This equation X that we've found is the horizontal distance. So we can just plug in this value of tea into who are equation for X. And that was 1 22.9 right? Yes, 1 20 I have a 5.4 seconds, and that is going to take five or 600 63.66 feet. So that's the horizontal distance when it hits the ground. Finally, hard day. Oh, uh, another thing about what part see we will see is that, um, consistent with the graph. This 600 feet, man, Yeah, let's say it's close. Kind of pulled this down a little much, you know, if this was really a true probable, that would be something like that. So let's be 500 5600 6 63 Yeah, it's pretty close. That's just a rough sketch, and I say That's about about Rod. So 663.66 feet is where the ball is gonna hit the ground. And Part Day says, use the graft to describe something about the path of the baseball that might be of interest to the player who hit the ball so we won't know how. What. The baseball player. But I wouldn't think so. Let's Paul is a video here, and let's go look up and see what the average length of a baseball field is. Back in a sec. Okay, so we're back now, and I found some interesting information that we might I want to tell our baseball player about. So let's look at Fenway Park, which is the home of the Boston Red Sox. If you're familiar of baseball, this is the green monster out in left field. They have this really huge wall. Let's go green because it's the Green Monster. It's really big tone wall at it. Left field this 37.2 feet tall, and the reason it's so tall is because Fenway Park Field is actually a little bit shorter than most fields. It's only from home plate to the left field wall is only 310 feet, and that's still pretty good ways. And I know I could hit a ball over the fence, but I guess these MLB sluggers probably good. So they made the war really tall. Compensate for this relatively short distance. No, this baseball player. With these initial conditions, let's see if this baseball player would have hit a home run, so we won't know what's the hot that 300 10 feet and happen. We find that out. Let's see well, at X equals 310 feet. What would be t? So we've got an equation for X X equals 1 22.9 roughly empty. So we're gonna antennas x 1 22.90 That's gonna tell us that T is equal to you. 3 10 divided by 1 22 or not. That's two point 52 seconds. So that means if this ball were to go over the fence, it would cross the top of the fence. You would say like this. And across the top of the fence. 2.52 seconds after it left the bat. Well, 2.52 seconds. How high is the ball and we can. We've got an equation for why they get us the height of the balls. Let's go back up here. Copy that back down. Why is equal to three plus 1 50 son, 135 degrees times t 16 t squared that we'll check that. Yeah. So now let's believe this value of tea into this equation for a while to see how the ball was at that time. 50 son 1 35 degrees times 2452 seconds. My 16 times 2.52 seconds squared, and I'll pause again and we'll see what that is. I have to run through the calculator. So after running this through the calculator, I've got some good news for our baseball player, because at that particular instance, in time, our baseball is 168.7 feet above the ground. And so, with our green monster being only 37 point to you, the that's definitely greater than 37.2 feet and good news to our baseball player. He has made a oh broad open door. This problem. I think this is a top, a problem that's really big for people who were looking to pay his assistant. Engineers enjoy this type of problem. May be looking at that profession because I'm an engineering student and I really enjoyed doing problems like this. I hope you did too.