All right, So we have a projectile and I'll just start with me and the information that's given to us. We know that the initial position, uh, of our projectile, I want to say r of zero. It's at the origin. Just have it at the origin. At the beginning, both X and Y equals zero, and we know that the initial velocity top place right here you not equals. It's wealth used for second in the eye direction and plus 49 49 meters per second in the direction That's the initial velocity and that. So you have both your X and your Y component right off the bat from this philosophy it's given to us. We also know other important fat, productive emotion. Your ex initial is going to be equal to the X component of velocity the entire time. That won't change, which is a convenient thing about project Ah, motion and then recording are why component of velocity. This is going to be equal to the initial velocity which we actually have conveniently right here. The initial velocity and minus we're fighting gravity with the little minus G t all right, and the X component of the position we can find that very conveniently. Um, by doing one of the first equation is you learned cinematics. X equals velocity in the X direction times time, the white component, the position of we didn't find it By using one of our very old faithful can imagine equations lossy in the Y direction times time. My guess is defying gravity, my ass. 1/2 g t sward all right. And we can find the projectiles distance Or don't distance Manitou that by using just good old with a gris If you just take your components, you square them both. You can find that radio distance over time. So now we have to do is substitute values into all these equations. And when you do that, you can get a ton of different values and you can make a table, which is what we did here. Gonna pull that out here. So take a look at that very dope, so make a table. And here you can see we have our time from 0 to 10 seconds. Velocity in the X direction. You can see states constant the whole time. Velocity in the Y direction beginning positive and then you can see it becomes negative after our reaches the max Ah heights. When you have your position over time in the X direction, which is stays positive the whole time, um, increasing from zero. And then for the y direction or the Y position, you see, it increased from zero gets to its max, heightened and decreases. And in the radio position, you can see conveniently matches. Um, the X motion. For the most part here you see crops all the way across and ends at the 1 20 as well. And that's just a vector that is essentially tracking whole motion. So it's not gonna be the same as the X, but it's gonna have that same starting value and ending value as the X that you can see there with zero on the top of the 1 20 on the bottom. All right, now we want to also consider something here. We want to consider the fact that the velocity is not perpendicular to the displacement where the distances Maxwell. So there's only gonna be some component. The velocity along are opposite to the direction of this radio factor. This component of velocity will either decrease or increase the value off our magnitude of our radio actor. It cannot be maximum. So this means the velocity back. You must be perpendicular to our that actually don't like this. The velocity vector has to be perpendicular to our that's perpendicular to our, um, when the magnitude of the radio component is maximum is Max important, important fact everything to work with. All right, so now at this point, we're going to starting with what we're given here. At the beginning, we had our initial velocity and we had our acceleration, which we know is right. My acceleration here cell ratio equals minus nine point each one meters per second squared. And we want to consider the magnitude of displacement vector. We maximum at time t. So let's I use this year to describe Sparks is out of here, and we'll do velocity equals V nuts plus acceleration, times, time and we can plug in what we know for this. We already know what the initial velocity is, and we know what the acceleration is. We just have to make sure we put a minus in for the acceleration, and then we're gonna be able to come have just time left over So we will get G equals plugging in Vina here this factor first and then plugging in the acceleration second and cleaning it up. That's why we get 12.0, in the direction plus using my Prentice easier. Right? But we have 49 0.0, in the J direction minus the 9.1 meters per second square times t also in the direction. So that's why we have these guys couldn't hear together And we can also get a displacement vector are doing a peanut T plus 1/2 eighties word. And when you plug, you know all the stuff that we have that we have the initial velocity vector right here and we also know that the acceleration is the negative 0.4 meters per second squared. So when you plug all that in and you go ahead and clean this up, you look at something of you sure are equals. Well, you know, t I had waas 49. Well, which is that j 60 minus the four point on which is part of 1/2 of the acceleration 4.9 xi squared. Well, this isn't a J direction now maximum displacement. The velocity in the position vectors are perpendicular. Hence, the doc product will be zero very important fact here, the the or at maximum displacement, the velocity and position vectors here perpendicular. So these guys zero doc wallet. So when I move this over here in the game, this up here is not years 12 I waas or 49 minus 9.1 T. And, of course, we have his full thing with just boxes in right here. It started dark products with our radio vector, which we found here really running out of space there. And this is, of course, in the J direction. This last guy. And then when you run through this Doc Products, um, you run through it, find that equals zero seconds, 5.69 seconds and 9.3 seconds. Lovely around the displacement at zero seconds in 9.3 seconds, um, are actually the minimum. And so the maximum displacement occurs. Actually, just had this middle one here is 5.69 seconds. All right, so we want to substitute this value of tea into the equation of displacement, which is this equation right here. And what I'm gonna do much. You gonna take my, uh, my lowly table and witness to the sides? We can actually have soul more space here, you know, right there. All right, so we're gonna take our our equation here, or are, And we can now plug in the 5.69 seconds and each one. He's my direction. Plus the 49 you know, times 5.69 minus four point times 4.69 squared. All right. And then when you do this, you don't figure out both components, which is 68.3 and direction and 120.2 in the J direction. If you want to find the magnitude, we will use the same thing over here. What we've that before. So the magnitude of the radio vector being 68.3, I swear, plus 320.2 squared. And then you want to get roots of this, and you're gonna gets roughly 100 on 38 years now just to kind of build long. We just did here. I had to use the fact that for maximum displacement, velocity, vector and the displacement vectors of perpendicular. That's always what we had that dot product that fee dot are down here is we have to use that fact about the fourth organ alley essentially with other than the perpendicular at those specific points now of two vectors of perpendicular. You know that that part of zero So first I had to consider that time t the displacements, baby Maxwell. Um And then I calculate the displacement in the velocity of tea. And then I took stock product and said the dot party with zero since they are gonna be perpendicular and will this solve for time and in knowing time you're employed into our radio expression and, um, find both components you x and the Y components, and then go and find the magnitude here, which is what we did.