https://youtu.be/Pmz3c6YYhp0
PHYS 1101: Lecture Seven, Part Six
So, here’s a picture to help me summarize for you, the key vectors that we’re going to work with in 2D. To really help define the scope of the typical problem that we’ll have. We’re going to be describing the motion. For example, here’s a car that’s rounding a corner. They’ll be some initial snapshot in time, put a spot here for the car being at this location at time t-0. And then, at the end of our problem, I’m going to put another spot here to indicate at Time t. This is the position of this object whose motion we want to describe.
Here are the key variables that we need, the key vectors. We’re going to want position vectors at the start of the problem, at t-0 and at the end of the problem, at Time t. So, somewhere, we’ve defined an origin. We have to know where 0-0 is. And then, these position vectors describe the coordinates, the x-0, y-0 at the initial time, t-0. The coordinates x, y at the final time, at time-t, the end of the problem.
Between these two times, we may have reason to think about the displacement vector. The displacement vector is the vector difference between r-0 and r. From start to finish, it’s the arrow that points from where we were to where we ended up. That’s Delta r.
And then, we may have need to talk about the average velocity. Velocity, remember, is displacement over time. So, in two dimensions, it becomes this vector Delta r over Delta t. If these two times were just two brief snapshots in a motion diagram, we’re accustomed to drawing between these two spots, a velocity vector. That would be this, an average velocity. It’s the same direction as the displacement between these two points.
We’re going to need, then, the next level, which is the acceleration vector. Is the difference between two velocity vectors divided by time. So, each of these vectors; position, velocity and acceleration, we’re going to have to work with the scalar components of those.
For this one, I have got x knot and y knot. For this vector, I have the scalar components x and y. Here they are, x and y. For velocity, I’m going to have an initial velocity in my problem and a final velocity. Those are going to have x and y components. And then, I’m going to have an acceleration.
So, here’s a good sketch of that. Somewhere, I’ve got my origin. I’ve got the scope of my problem. The motion I’m focused on from start to finish. Start is going to mean at Time t-0, where was the object? And usually, I think I can say, always, we’re going to start the clock then. Motion is going to occur and then, at some later time, t, that object is going to be at this location. With its final coordinates x-y, given that it had some initial coordinates, x-0, y-0.
Again, x-0 is going to mean the literal value compared to the origin of where it’s sitting above the x-axis. And it’s y-0 is going to be where it’s sitting above the y-axis or along the y-axis. And then, the same idea here at the end. What’s the x-y coordinates at this instant?
At the beginning of the problem, I can have an initial velocity. That literally is the speed and the direction that that object is headed right here. I’m going to have to break that vector up into a v-0 y component. And a v-0 x component to describe this motion.
So, a real life hypotenuse, I’m going to work with these two sides of this right triangle. Likewise, at the end of the problem, my final velocity, I’m going to need to work with the x-component and the y-component. Those vectors.
My acceleration may have a y-component and an x. Okay, it’s a lot of variables, now, so if you had a hard time remembering the 7 that we dealt with in Chapter 2, you now are going to be taxed further and have to remember more variables. But hopefully, you understand now, the importance of doing that. You have to know when I show you the variable v-0 x, that that physically has to represent the horizontal part of the initial velocity.
The velocity of that object at t-0. V-y, it has to represent at Time-t, the end of our problem, the vertical component of the total, final velocity vector. Hopefully you get the idea. Memorize those. Make yourself flashcards.
Here, I just have it sketched again for you to emphasize this point. I’m not going to belabor it. But everything I have in blue is going to represent variables that are connected and relate to the vertical part to the motion. I’ve got a y-0, an initial y part to the velocity. A-y, final v-y.
In purple, I’ve got the variables that describe the horizontal part to the motion. X-knot, e-0-x, a-x, etc. These variables, let’s count them. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 12 of them, now, that you have to memorize and know physically what they mean. It’s time, position and velocity at the start. Time, position, velocity at the end. And the acceleration in-between.
So, here is quiz question for you, number 7. I’ll just let you pause the video here and read it yourself. How many variables do we need now? And let’s include t-0, even though we’re always going to set that equal to 0. How many mathematical variables are we going to have to work with now, to uniquely describe motion in two dimensions?