https://youtu.be/gNM8m5eXQnE
PHYS 1101: Lecture Eight, Part Two
So where were we? Here’s my summary from last lecture. We spent a lot of time looking at curved trajectories in 2D, and looking at how to dissect them, or analyze them. The first point, or bullet, was the notion that in 2D we can have curved trajectories, and this will be the case whenever an initial velocity of some object experiences an acceleration that is not parallel, or anti-parallel to this velocity.
Any delta-v that’s got to be added to every subsequent velocity pointing in a different direction is going to cause the velocity to curve, and I walked you through three examples. The next main point, and I emphasized this in many different ways, is the idea that to do the mathematics, to solve these problems we’re going to have to take the real life picture of the motion, which is going to involve appreciating what position vectors are initially, and at the end the x coordinate, the y coordinates, the velocities at the beginning and the end, and the acceleration vector, that is constant vector throughout the motion.
Appreciate what these are in real life for the problem, but then to apply the mathematics, we’re going to have to break every one of these vectors up into components, and you do that because the mathematics then becomes simpler. We can leverage the fact that all horizontal components, or x components of all these vectors are the only vectors that interact, or influence each other. So it’s a horizontal velocity component that can only change the horizontal position. It’s only a horizontal acceleration component, a delta-vx that can change Vx.
Likewise, for the vertical components we have the same relationship. Only a Vy component causes the y coordinate of an objects position to change. Only a y component to the acceleration will cause the y component of the velocity to change. By separating it out in this way, we have turned what is a complicated curved trajectory motion into something into something that we mathematically can handle well. We can treat it as one dimensional problems.
One of the dimensions is all the horizontal vectors, and the other is the vertical vector components, and the mathematics then becomes straightforward because once we’re restricted to one dimension, we can keep track of the direction of these vectors mathematically with a sign, and that’s what makes our math work out. It makes the addition and subtraction of these terms work to predict what the velocity will be later, or what the position will be later. In real life this is the picture we’re going to have. There’s going to be some object at an initial time in some location.
Because it’s headed in a certain direction with some speed, and it is undergoing some acceleration, the delta-v vector is at some angle, specific direction, it’s going to cause this velocity to change, and the object will follow some curved path. At a later time, it’s going to end up at a different position, a different x coordinate, and y coordinate and a different velocity. These are the real life entities that your eye is drawn to, the initial speed, and the direction the object starts out heading, a real…the total acceleration vector, and then at the end of the problem, how fast is it going, this magnitude, and then what direction is it headed at that instant.
This is the real life picture, and now I’m going to show you what we need to focus on to solve the problem. And this is the real life picture. This is where we start when we read a problem. Then we’re going to have to go from this picture to the picture that focuses on the components of each vector. So notice now instead of emphasizing the speed, and the initial direction, the resultant velocity vector initially, I now have in bold the components of that vector.
There’s the y and the x. Notation gets cumbersome once we go into 2D. This V0y, the 0 there indicates it’s the initial velocity. The y indicates it’s the y component, only the vertical part. And similarly you’ll see this trend with the other subscripts that you use.
Okay, so what are our 1D variables that are interacting? Initial x position, the initial horizontal part to the velocity, the Ax, the part of the acceleration that will cause the horizontal velocity to change, and the result of that, as time goes on, is my position. Sorry about that. My position is now my x coordinate has changed, and my final velocity, horizontally, has changed.
This is the delta-vx that I add to this initial x velocity vector every second that leaves me, at the end of the problem, with the final Vx. Let me grab green. My other one dimensional variables then is the y. I have an initial y position, an initial vertical part to my y velocity, an…Excuse me, an initial vertical part to my velocity. I have the part of the acceleration vector in the vertical direction, which is what causes this vertical part of velocity to change. The result is at some later time I end up at a different position, a different y coordinate, and my y component of the final velocity has changed.
This is the delta-v that changes this vector every second to become this vector by the end. So that’s the language, and then the description to do the math, to solve the problems, and then I’m going to show you that with examples in this lecture.
As usual, we’re going to start out new material with a couple of reading quiz questions. These questions relate of course, to this section that we’re focusing on in this Lecture 3.2.
Question 2: It is important to realize that the x part of the motion occurs exactly as it would if the y part did not occur at all. Is that true, or is that false? Just fix that typo for you.
Next question: In the equations for kinematics in 2D, the variable v sub y will always represent… I’ll let you pause here and read these choices.