Hey guys, so that's probably have a remote control car that is subject to a force that varies over some distance here. So we have these forced distance graphs, we're gonna take the area under the curve, and that's gonna be the work we want to do is we want to calculate the speed of the card excuse for now. The car initially starts from rest, so this is the not equal zero. So we want to do is we want to calculate the final speed. So how do we do that? What we're going to calculate, we're going to relate the work that's done to the change in the kinetic energy by using the work energy theorem, the network is equal to the change in the kinetic energy. So what I'm gonna do is I'm gonna have to take the area under the curve for this whole entire graph here, up until X equals four. And then I'm gonna relate that to K final minus K. Initial. So what I'm gonna do is I'm gonna break this up to do a couple of sections. The work done from 0 to 2 plus the work done from 2 to 3 plus the work done by three. From 3 to 4. That is going to be equal to the K final minus K. Initial. However, because the initial speed is equal to zero, there is no initial kinetic energy and all of this is going to be final. All this is gonna be final kinetic energy, which then I can relate to the speed of the car. So what I can do here is I can take a look at the areas under the curves for each one of the three little terms that I've I've made. So the work done from 0 to 2 is basically going to be all of this distance for all of this area right here. So I'm gonna highlight this some blue, that's going to be this guy. Now, what about from 2 to 3? Well, 2 to 3, there actually is no area under the curve because we're at zero, there is no force acting on this car from 2 to 3. So there is gonna be no work done. And then finally from 3 to 4 there's definitely gonna be some work done because it's gonna be this area here under the graph. So I've got those highlighted areas here. Let's go ahead and start calculating. So if I'm gonna calculate the work done from 0-2, basically what I'm gonna do is I'm gonna cut this up into two shapes. I've got one triangle and one rectangle. I'm gonna do this all in one sort of line. But basically this is gonna come down to two terms. I've got the area of a triangle which is one half the base, this is gonna be one and the height of this is actually going to be 20. So I've got one half of one times 20. And then plus this area over here which is really just a rectangle, it's just gonna be base times height, so the base is one and the height is 20. So if you go ahead and work this out, what you're gonna get is you're gonna get 30 jewels. So 30 jewels is the first little section of this right here. So actually going to highlight this in blue. So let's go ahead. Now calculate the work done from 3-4. So the worked over 34 is really just gonna be this little triangle right here. And so I've got is one half of one that's the base and then the height of this thing. Well this doesn't actually hit all the way down to uh to negative 20 and it's a little bit past negative 10. So I'm gonna do is I'm gonna call this negative 15 years right in between, remember it's negative because we're actually in the UAE and the negative y axis here. So the work that's done is actually gonna be negative 7.5 jewels and that is the work done from 3 to 4. So now I really just add these two things together. And so what really, this is just becomes, is I have 30 jewels Plus this is negative 7.5 jewels And this is equal to one half of em. So we actually know what the masses, this is actually gonna be four times v final squared here. So this is the final squared, so I get 22.5 jewels equals, this is going to be to the final squared. So you go ahead and work this out what you're gonna get. You're gonna get two points. Let's see here. Against 3.35, you get 3.35 m per second. So that's the final answer here. All right, So that's that's it for this one. That's the speed of the car at X equals four. Let me let me know if you guys have any questions.