 Now let's introduce constant acceleration in two dimensions. When we had one dimension constant acceleration, we had a couple equations that we were able to derive from calculus. We had three additional equations down here that we were able to derive using the definition of constant acceleration in terms of the velocity average, as well as a couple that came just from algebra. In total, between all these equations, we had five variables, the change in position, initial final velocity, acceleration, and time. And because we were only working in one dimension, that change in position was something like delta x. If it was a vertical motion, this would be delta y. But in either case, our velocities and our accelerations were limited to whichever dimension it was we talked about here, either horizontal or vertical. When I have two dimensions, my equations that come from calculus become vector equations. So it's still my change in position, but now it's the vector change in position. And I still have my initial velocity, but it's my vector initial velocity. So I still have five variables, but four of those variables are now vectors. Remember, time isn't a vector, it doesn't have a direction, it's just how much time has passed. Because these are vector quantities, I'm doing things like vector addition and multiplying a vector by a scalar. And the math on vector equations wasn't just as simple as adding the two numbers, direction mattered very, very much. You remember, though, that when we did vector math, if we could get our vectors into component forms, it made the math a lot easier. One of the things that we saw if we had components was if I have two vectors that are supposed to be equal, the components are also equal. So for example, if a and b vectors were both equal to each other, then their components had to be equal to each other. The x components and the y components each had to balance out so that the a and b components were exactly the same. Well, if I apply that type of concept to one of my vector math equations here, what I see is that the components in x on this side has to be balanced out by the x components on this side. So that works out that I can express the horizontal aspect of this equation in terms of the horizontal displacement, horizontal initial velocity, and horizontal acceleration. Similarly, I have the equation for the vertical direction with delta y, vy, i, and ay. If I apply this principle to all five of my equations for constant acceleration, I get one set of five equations for the x and a similar sort of equations for the y components. So I now have 10 equations that I'm working with. Similarly, if I want to use the strategy of listing all my variables and keeping track of them, I now have two lists, one list for the x and one list for the y side. And remember that time isn't a variable, so there's no x component of the time and y component of the time. Time connects these two sides together. So when I'm solving them, I have a rule of three plus. I still have the fact that if I have three known values on one side, either the x side or on the y side, that I can solve for the unknowns on that side. But I also have that once I know the time on either side, I automatically use the same value for time on the other side. So that's going to help me solve some of these problems as well. Remember that vectors don't have to be given in component form. They're often given to you in polar form. For example, if you've got an initial speed, that's really the magnitude of your initial velocity. So if I have an initial speed of 25 meters per second, that's my magnitude. And if it's tugged that that's at an angle of 30 degrees above horizontal, that covers my direction. Well, then I could use trig to find the individual components, vix and viy, using my normal trig rules. And you might want to go back and recalculate these to check for yourself to make sure it's working the way you think it is. In total, that's our introduction to constant acceleration in two dimension, just like what we were doing in one dimension, except for now I do it for x and for y at the same time with time connecting the two sides