https://youtu.be/kdw-iEYu5qM
PHYS 1101: Lecture Ten, Part Two
The “Where were we?” I want to cover today gives an excellent overview, I think, of what we’ve done so far in this class. Beyond Chapter 1, where we developed some mathematical tools and skills that we’ll need, we began understanding motion by fine tuning our view of it and appreciating that we want to identify, pick out, see from motion three basic characteristics. At any instant, we want to think about the position of that object, the instantaneous velocity, the meters per second, how fast it’s going, and the direction at that instant. Then also, at that instant, the acceleration. Is this object speeding up? Slowing down? How is this velocity changing? I want you to think of these three things as fundamental building blocks that uniquely capture describe motion. Position, velocity, and acceleration.
With those three generic building blocks, we know, just given specific values of them, we could describe all kinds of motion, different scenarios, whether it be an object that’s stationary, zero velocity, zero acceleration, but it has a specific position coordinate, given an origin that gives meaning to that number. An object that’s moving along at a constant velocity will have a value for v, acceleration of zero, and then at any instant, I could talk about its position coordinate. Then, for acceleration, I only have meaning to specific instantaneous velocities as this object speeds up, and a specific constant acceleration that tells me, every second, by how much is the velocity changing. How much is it being stretched or compressed, so to speak, for 1D motion.
So, that’s pretty powerful. With these three building blocks, I can apply those to all kinds of motion. We then went onto realize that by thinking of these building blocks in terms of a scope of a problem, where we want to understand the motion from some initial instant as the clock is ticking, and we watch it to a final instant, that if we apply the ideas of position and velocity at the start, the unique position and velocity at the end, then as long as I have a constant or zero acceleration between those two instances, I have a powerful set of equations that I can use to learn a lot about the motion, to predict what the initial velocity must have been, or the initial speed, or the final position.
So, here, again, were the criteria that I had to satisfy in order to be able to solve problems using this technique. It only works focusing on a single object whose position and velocity are defined by those variables. You have to keep in mind that during the scope of the problem, from when the clock starts to when the clock finishes, I have to have a constant “a” or a constant change in velocity, a little delta-v vector, that’s the same every second during that whole duration of the problem.
With those building blocks, we have these very useful equations that we can apply to that problem to solve for the motion and to learn about it. These equations are nothing more than a mathematical logical consequence of simply the definition of these fundamental building blocks.
For example, the first equation tells you that the later velocity is just the initial velocity. But then, I have to add to it acceleration times time. Acceleration times time just tells me if, for example, 5 seconds have gone by, and my acceleration is, let’s take this 3 meters per second, then I know after 5 seconds, my velocity has changed by 5 times 3, or it’s now changed by 15 meters per second. So, my later velocity then is what it was initially, and then I’ve changed it by the 15 meters per second, for example.
So, these are just a logical consequence of the definition of position, velocity being the change in position divided by time, and acceleration being the change in velocity divided by time.
Okay. The set that I show you here is the horizontal components. So, this is the set that we must use when working with a two-dimensional problem. That means that the motion is following a curved arc, a curved trajectory. Of course, we know we have a similar set of equations that we have to work with that captures the vertical part to the motion. For those curved trajectories in 2D, that horizontal part and the vertical part, they’re really separate parts of the motion. The only trick is, or feature is that they’re occurring over the same time as the clock is ticking.
Okay. This whole issue, these exercises that we’ve done in Chapter 2 and 3, is called kinematics. That just means we’re simply describing the motion. We’re not asking how we got the acceleration. We’re just saying, “Given that it is what it is, these equations govern how those terms have to logically be related.” We can leverage that to describe the motion, to solve for it, to tell you where the object ends up, how fast it was going.
Okay. Here’s a critical bullet I want you to think about as you prepare for Exam 1. You need to be sure on this exam, going into it, that you know what 1D motion is versus 2D motion. Your criteria is you need to picture the motion, and if that motion diagram you would draw follows a straight line, the angle of the line is not important, just that you could lay a ruler down and draw a straight line through those dots. You’re working with one-dimensional motion. With one-dimensional motion then, you only have to work with one set of equations. You only have a position, a velocity, and acceleration all along that line. Call that the x axis, or if it’s vertical motion, perhaps you want to use y.
But, you don’t have to worry about breaking vectors up into components. You’re just going to use the sign of, say, velocity and acceleration, to capture the direction. If, however, you read this problem, you picture the motion, you realize that the velocity from the start to the end, the angle of that has changed, you know then that the trajectory of this object had to follow a curve between the start and finish during this constant acceleration.
If your final initial velocity are at different angles, you’re in two dimensions. In two dimensions, you’ve got to follow the problem solving steps for two-dimensional kinematics. These are more involved. Now, you’ve got to break all your fundamental vectors, velocity at the start, velocity at the end, acceleration. You’ve got to break those up into their horizontal and their vertical components. You got to do trig to do that.
These position vectors, you effectively are also breaking those up into components, just by noting what their initial coordinates are compared to an origin, x0 and y0, and their final coordinates compared to that same origin, x0 and y0. To start our new material, let’s do our two reading quiz questions. These come out of material from these sections out of your book.
Question 2, very important concept, difficult for people to really accept and embrace. In physics, what we mean by force, to put it in laymen’s terms, means what? Is it A, a push or pull? B, something that results from motion? C, the cause of motion to stop? None of the above. What is always true with respect to what we mean when we say “force” in physics?
Question 3: To solve problems with Newton’s Laws, we will have to break force vectors into their components. Is that true, false, or not in the assigned reading?