https://youtu.be/Nu2me6gIlQg
PHYS 1101: Lecture Eight, Part Three
Before we do our example, let me give you the big picture of where this work fits in. I remind you of my tree schematic, but I have shown you before in the context of our one dimensional problems that we worked. We start with the basic equations that define how these quantities have to be related.
For example, a final velocity is equal to the initial plus acceleration times time, etcetera. This first equation, let me just point out, is nothing more than the definition of acceleration. You can rearrange this equation to say acceleration is equal to v minus v0 divided by t. These are just a natural, logical consequence of the definition of what velocity, these position values, and acceleration mean.
With these generic equations at the base of our tree, we’ve worked on some skills and techniques for envisioning the motion that help us figure out what these variables represent and therefore how to use these equations to logically progress and customize these equations to a specific solution to a problem. A ball rolls straight down a hill. How far has it gone after two seconds, etcetera? Now we’re doing 2D and let me start out by erasing a few things for you. Again, we’re going to be focusing on problem solving steps to get us from the start to our set of problems. But now let me grab a different color here to emphasize that it’s really a new set of problems that we have the skills to solve. A new set of leaves, if you will, where each leaf represents a specific problem. Our goal is going to be, again, logically thinking through to arrive at the solution to a specific problem. But now it’s two dimensional kinematic problems.
Describing the motion of curved trajectories where that may be figuring out an initial speed given some path or some consequence or finding out the final position coordinates, etc. What we have up here at the top of our tree now or that, I’m sorry, really at the base of the tree at the very start of our problem solving strategy is going to be two sets of equations. There’s going to be a set for the x coordinates, the x components of all the vectors, and for the y. So to emphasize that, let me just go in and show you.
Here’s what we do. The same three sets of equations apply to x components of all these vectors. Now this isn’t just v. I’m going to put a subscript here to indicate this is the final x component of the velocity. And this is the initial x component of the velocity. This is the x component of the acceleration, etc. All these vectors, I’m going in and I’m just customizing them to emphasize or to remind myself that it’s only the components that apply. Same thing applies to equation three. I’m going to call these sets of equations in purple here my x variable or x component equations. The other set I’ve drawn, let’s change the color so it’s a little bit clearer, and I’m going to go in then with that dark blue pen to clearly indicate that now these are y components. And you know what, I’m going to change these position variables to be y and y sub zero. And let me put my subscripts in. Okay, just pause to tidy that up a little bit for you. And spread these equations out a little bit because it does become daunting to keep track of and to focus on all these subscripts.
Once you get past getting distracted by all these subscripts, you’ll see though that these are just the same three equations we’ve dealt with before. The first one involves the change in velocity, acceleration, and time. The next one has to do with the change in position resulting from an initial velocity, acceleration, and time. And equation three is useful if ever you’re not given the time or not interested in the time delay or the time change during this problem or during the motion. Same set of equations but those that relate to the x vector components and then the same set relating to y vector components.
Around all of this I could draw a box or a circle and say these are my starting equations for 2D kinematics. What’s going to go in here? That’s the same idea that I’ve shown you before. I’m going to go give you some problem solving steps to follow to help you go from these generic equations that apply to any two dimensional trajectory to customizing those to solve a specific leaf or a specific two dimensional problem.