Given the position vector minus. Uh well we're told basically that were actually given a scenario here right? We have a projectile shot vertically upwards from six ft off the ground. On the position which is his upward is given by -16. He squared plus 40 18 plus six. The velocity then is minus 32 T plus 48. So at T equals zero. We shot it up at a velocity of 48. They give us any units feet I guess feet per 2nd 48 ft per second. Um Yeah they gave us feet but they didn't really explicitly tell us that there are seconds. But Yeah it's going to be seconds because this is 32.2 G is about 32 and feet per second squared. So if this were inaccurate scenario on Earth then that would be the gravitational acceleration. So let's see here we they want to know let's see, we need to figure out when it's velocities lesson zero and greater than zero so we can figure out when the velocity equals zero And that we just saw for that. And we get T is 3/2ves wasn't he one called it? So uh 3/2 of a second at 1.5 seconds it's zero. And so if we look it's greater than zero between zero and Um 3/2 1.5. And then where did that get? How did I like to there three halves, it's less than zero. And as we go from 3 1.5 two whenever it hit the ground again, they call the capital P. Um We could figure that out but they didn't ask us to figure that out. Obviously we set this equal to zero and figure that out. Now we can see then they ask us for the maximum height and that's just the maximum height occurs when the velocity is zero. And so we can plug this back into here And we get that, that is 42 ft. So we go up 42 ft. Well we actually go up whatever 36 ft because we started out at six ft. So the maximum height is 42 ft above the ground. Now the next problem is though, this in their effort, the distance the body moves down an inclined plane. No, no, no that's that's not the right one. Particle moves positions measured, determine the distance travel. Oh right. Um I was gonna make a graph for this one. Let's see if I let's see if it's here. Let me add it in real time here. Thank um on connect to this area. Yeah, there it is. Yeah. So I added a graph there for you. So they want the total distance traveled. So you know, we basically, if we think about it, we started minus 10, go up to I think that's five. I didn't calculate yeah, that's five And then go back on down to four I believe. So, you know, we gotta add this distance and then this distance to. So the total distance isn't this right? That's just the displacement. The total distance is this plus this. All right. So we have to go from, we got to find the vote on the velocity zero and that occurs when The velocity is greater than zero between 1. We're talking the ranges between one and six, so the velocity is positive between one and five and it's negative between five and six. So we need to add the position how far we went and this region here. So that's s Evaluated at five -S evaluated at one and then how far we went here and that's as evaluated at five minus evaluated at six. So we wind up with putting all that, in we wind up with 17cm so that the total distance, we moved. the total distance traveled um Is 17 cm now. It only moved. Let's see. Uh Well you can figure that if the ending position, how far it is from the starting position, I'm not sure we could, we could figure that out, but it's less than this. Obviously, I think it's to think it's 16. I think this just drops by one here. Well, maybe it would be 15 and I think, yeah, so anyway, that's how you just got to be careful about, you know, seeing where the signs change. So you can kind of at a distance is whenever the velocity is negative, you want to basically change the sign of the distance that you're adding.