Hey guys, welcome back. So today we're going to be continuing our talk on 2D kinematic equations. We're going to start off with this animation here of a typical problem. You'll see a ball being kicked and what I want to point out here is that our velocity here is going to have both x and y components. Okay, so you have the v sub x equals v cosine theta with the theta being this angle here, v sub y being your v sine theta.
This is how we represent these components. And what I want you guys to watch out for as this plays is how at each point of motion, you have velocity components in both the y and the x directions the whole time. These 2D problems are much more similar to real-life situations than the 1D ones that we were talking about before.
So let's just review those real quick. We have displacement, right, with our 1D kinematics. It was our delta x equals x sub f minus x sub y.
i are final minus initial positions and then this was the backbone for all of these velocity speed and acceleration equations right so now let's look at a 2d situation okay so we have our graph here an initial point a final point we have these position vectors that come from the origin there right and they're represented here mathematically as x, x hat plus y, y hats, and these are unit vectors, okay? These things with the hats on them. Sometimes you'll see x and y as i and j, just depends on the context. But this vector here, this is going to be our new one, right?
This is the change in our r vector, our position vector. It's basically just this, rf minus ri. We'll see that here, this is our displacement, right? and you can also represent this as delta x, x hat plus delta y, y hat. So these two equations like we just talked about, these are the changes we're making from our 1D kinematic definitions to our 2D kinematic definitions.
Okay, so you see how before it was just our x components and the velocity too, for example. We have just our delta x over delta t. Well now when we want to talk about our average velocity, we've got to do...
delta x over delta t x hat plus delta y over delta t y hat. Okay, so we're adding in this y component, same here, down here, you see the subscript y terms, and with the average and instantaneous velocity as well as the acceleration. So these are just a review of the 1D kinematic equations that we have. We have a final velocity right here, final position right here, we have a final velocity squared right here, average velocity.
And keeping in mind, of course, that these are all with constant acceleration. So now we want to take these and we want to make them two-dimensional. So how do we do that?
Well, we have our three basic constant acceleration equations, right? So let's look at this first one, this purple one right here. What are we going to do? We're just going to separate them into their x and y components, okay? So we just have these two things.
Instead of delta v equals a t, we have delta vx equals axt, all right? And the same thing, delta vy equals ayt. just the components. We're going to do the exact same thing for the green and blue equations right here. You'll see that we have x terms, y terms, delta x, delta y, same thing over here, all right?
And then our fourth standard equation, we're only going to use in component form, so we would say this is the x component of vf squared and the y component of vf squared. Okay guys, so you have a problem, projectile motion. This graph is just like the animation at the beginning.
One object's moving, but you have two-dimensional quantities, right? So you have components in your x and y for position. You have x and y components in your velocity moving here.
You have x and y components in your acceleration. Here it's constantly decelerating. So you get this problem.
What do you want to do? First thing is you should look at your x and your y motion as completely separate. It'll help you understand the problem better.
You're going to want to... break up those vectors, you're going to want to make versions of those general equations like before that are specific to your problem to help you figure out the unknowns. Next thing you're going to want to do is determine important events. So you're going to want to define these based on time, location, or maybe some special points in the motion.
An example, we can go back to our graph here, right? Maybe here at the beginning, this is pretty important. Your initial time, t equals 0 is here, your initial velocity is here. Maybe you can also note where your object ends up, right?
You're back in the same y position as before in this example here, your t is your final time, and those special places, you know, maybe the top of this motion path, that could be something important to jot down. So once you have these events picked out, you're going to want to make a chart like you guys have been doing in class, right? I'm going to make your columns these events, okay?
These are going to be again based off of the time. that these events are happening at. And over in your rows, you're going to organize, well, what do you know about these times? You know the horizontal position velocity and acceleration.
You also know your vertical position velocity and acceleration. And as you do the math and you figure out more information, you put them into your chart. That's going to help you organize everything a lot better. So just an overview, some tips.
You're going to treat your x and y motion independently. You're going to determine those important events. And then you're going to make a table of values for each event. Alright, so good luck and I'll see you guys next time.