https://youtu.be/7zLFppgiEag
PHYS 1101: Lecture Four, Part Two
As usual, before we start our new lecture content, I want to give you the big picture view, or summary, of our last lecture. Where were we last time? We finished up Chapter 1, and we learned really a key feature. After going over the vector description or the component description of a vector, we then looked into adding vectors by components. So, let me say a few words or remind you about the breaking vectors into components.
We’re going to work with a lot of quantities for which we need a vector description, a value and a direction. We’re going to use this arrow to represent these vectors. In real life, the values for these and the direction is what our eye is drawn to. It’s what we see. It’s what our speedometer reads, etc.
We’re going to, though, use a mathematical tool of breaking a vector up into its components in order to leverage the mathematics and solve the problem in physics. So, a vector is broken up into its components. As you can see the tail-to-tip addition, if I take Ay plus the vector Ax, that sum is the vector A from the initial tail to the final tip. So, we could write that as the vector A is equivalent to or is equal to the vector Ax plus Ay, or as I’ve drawn equivalently over here, another way to write this is that the vector A is equal to the vector Ay plus Ax.
These, when written this way, this entire quantity, these vectors where I’ve got the arrow over the whole quantity, those are called the vector components. What we’re going to work with though, most of the time, is what we call the scaler components. We can do this because we’re going to be breaking all of these vectors down into components that are all horizontal and the components that are all vertical.
Once they’re all aligned along a specific axis, then the direction we can easily keep track of by just using a sign. So, we rewrite this vector. An equivalent way of writing this is to say vector A is equal to just a number, with a sign in front of it, a scaler, times these unit vectors. So, this is equivalent to that. We’re just making a substitution.
But now, the sign here, as we now have learned in manipulating vectors, the sign will flip the direction of our little unit vector. The scaler number here stretches this unit vector, or multiplies it. This starts out with a value of 1. So, that’s why this becomes equivalent to this vector, captures the direction and the proper size. Working with the scaler, it’s nice, the scaler component, because it’s just a number with this sign in front. We’ve done a lot now, realizing that trigonometry is a great mathematical tool to help us get the lengths of the sides of these triangles to get the geometry straight, which then gives us the magnitude of these vectors.
We’re then going to pick the proper sign, based on the direction of these arrows. The vertical one has to capture the up/down trend of this resultant. The horizontal side has to capture the right/left trend.
The other summary from the previous lecture was using these scaler components to do more exact vector addition. We have our tail-to-tip graphical way of adding, say, vector A plus B to get us vector C, but you need more specific numbers beyond what you could do just with a ruler, given like a legend on a map that this length represents a certain amount, then figure out what this distance represents. We can get an exact number by doing our trigonometry, and then coming up with the scaler components for this resultant directly by adding together the similar scaler components of the vectors that make it up.
Best way to do this is to make yourself a table. I’ve got a column here for x components, a column for y. For every vector, I go in, vector A, and I do my trigonometry. I draw my right triangle. I find, with my trig, the scaler components for A. I do that for each vector, vector A and vector B. Those scaler components, paying attention to the sign, I’m going to put in here my scaler for x and for y for the first vector, do my trigonometry for vector B, put in the scaler component for x, the scaler component for y. Then, the sum of A plus B is just going to be adding down these columns, as I’ve got indicated here with the green arrow. What you end up with then, is that the sum of Ax plus Bx is the scaler component for C, in the x direction, and likewise for y.
Okay. You’ll practice one of those on your homework. The last topic we covered last lecture, we’re going to see many times, and even in today’s lecture, I’ll show you more examples of it. But, it’s this idea of a motion diagram. This really helps us to slow down and be able to pull out these key features of the motion that we need to focus on. These motion diagrams are going to help us be able to identify these new terms we’re going to define today.
The dots here would represent a time history of where the object was. Key to a motion diagram is that the time interval between these spots is the same. I’m going to call that Δt. I’ll define that, in fact, later in this lecture, but that just means a small time or time interval. The key is between all of these dots, you have the same time interval.
Okay. We learned on a motion diagram, once you’ve created this history by the location of these dots representing the object, you could either go in and add numbers above these dots to reflect the sequence in time, or a more useful treatment is to add a green arrow between all of the dots, or add an arrow. The direction of this arrow naturally tells you the time sequence, because it reflects the direction the object was headed. So, this had to have occurred earlier in time than that. We know that the spacing between dots, which becomes the length of this arrow, is a good representation of how fast the object is going. These arrows, we’ll learn, are a good visual of the velocity vector for this object.
A major key to this class is going to be realizing when an object is experiencing acceleration or not. These motion diagrams will really help you see that. If this velocity vector, as time goes on, changes in any way, gets longer, shorter, we’ll even learn that changing direction is important, that we have acceleration. Knowing that, being able to identify that, will be important.
Let me quickly show you some animations of different motion diagrams. Actually in this case, not animations, but just show you some different motion diagrams to give you a feeling for the different kinds of representations that you may come across, and how you can describe really any motion with this. Initially, we’re going to restrict our motion to one dimension, motion along the lines. So, let me show you some different examples of one dimensional motion diagrams.
These two show you the classic difference between an object moving at a constant velocity versus accelerating or speeding up. For all of these, let’s assume that the object moved from left to right. So, our velocity vectors between all these dots would point to the right. So, the obvious distinction between the two here is that the spacing is getting increasingly larger. This object is speeding up as time goes on. Those velocity vectors, if you were to draw them in, are getting larger. I have acceleration here. Here, velocity vector is staying the same, constant velocity, no acceleration.
Let’s compare these two briefly. You can see the velocity vector is getting larger for both, but let me point out. Obviously, you see a difference between the two, that I have more dots here than I have here. So, the spacing is larger. The difference here is that the acceleration, the amount that the velocity changes between each snapshot is more in this bottom example than the top. We’ll learn more about that today. So, I have acceleration here and here, but I see there’s a difference, that I have more. I can even recognize how much acceleration I have from these motion diagrams.
Let’s go down to a couple of these, and let me just walk you through what kind of motion this could represent. Again, we’re going to assume going from left to right. Here’s an example where the object, it’s going pretty fast, but then this spacing is getting closer. So, the object, somebody or something is slowing down.
Here, I see a whole cluster of points at the same location. The object must have stopped here for a while, and then started to the right again. This motion to the right, the spacing again now is getting larger. So, the object must be accelerating.
Do another quick example here. This object looks like it starts out moving at a constant velocity. When it reaches this position, it starts speeding up. I have acceleration. The velocity is getting larger.
Let’s see this example. I got several things going on. It looks like the object starts out at a constant velocity, slower than this object started out. Notice. Going along slower. Then, I go through an interval where this object slows down, looks like it comes to a stop again, and perhaps even wanders around with a step here or there, aimlessly for a while. Then, the object starts moving again very slowly at a constant velocity, slower than initially, and definitely slower than this object.
I just throw those examples out there, give you a feel for the kinds of motion diagrams you’ll see, and the strength of this tool in letting us just visually pick out velocity and whether it’s accelerating, speeding up, or slowing down.
Back to our lecture now. This section is our usual reading quiz. Lecture 4 is about these three sections. Hopefully, you read these before the lecture. The next two lecture quiz questions, number 2 and 3, have to do with these sections. These have to do with the specific variables that we’re going to use to represent certain physical quantities.
The first one has to do with the vector v0. In one dimension, we’re going to work with the scaler component of that description of that, which we would write just as v0. What is that going to represent? In question three, I’m asking about the variable x, and what’s that physically going to represent. It will represent these things, all of these variables, for every problem that we do. So, you have to memorize what these physically mean.
I ended last lecture with this quiz question for you, asking your opinion, when you just think about fundamental ideas of motion, what was most important? What the difference is between all three of these scenarios, one and two, or two and three. What I’ve emphasized a few times now, I’ll say it again, it’s not obvious when students first come into this class that acceleration is really an important feature of motion that we need to learn to recognize.
This woman is accelerating. Her velocity is increasing, and that motion, that characteristic, makes it distinctly different from either of these two, one or two. The key to that is seeing something specific and recognizing velocity, and then recognizing that that velocity is changing. As time goes on, the object is slowing down. It’s speeding up. Or, we’ll learn even the direction of the motion means I have acceleration. So, here’s my bullet statement yet again, that recognizing this acceleration is important in this class.
Now, we’re ready to start Chapter 2. What we have to do here, as I say, is go through these specific definitions of terms that are going to allow us to mathematically describe motion in a way that makes it very unique and very specific. We have to learn methods and terms for describing the location of an object at an instant in time, how fast it’s going at that instant and in what direction, and then does it have acceleration. Is it speeding up or slowing down? For every one of these, we’re going to need vectors to do that job, to make that information that we write down for location, for how fast and direction, to make that information specific enough.
So, let’s start with a general statement that applies to all three of these quantities that we need to define. Because this information has to do, and we’re going to need a vector to describe it, in order to give meaning to the sign and, in some cases, the value, you’ll see that we assign to these quantities, we have to have an axis and an origin defined. Otherwise, these quantities don’t have a unique meaning, and they’re not useful to us.
So, for all of these problems, the very first step is to define an axis and an origin. The axis is just going to be the direction along which these vectors are going to either be parallel or anti-parallel, and the origin is key. That’s going to define the position that we’re going to call 0. This is just a mathematical tool that we need in order to make our vector mathematical tools for the motion unique and to work out for us to solve the problem.
Where you define your origin and the sign that you choose to represent positive direction versus negative, those are just mathematical tools that you’re using to solve the problem. They can’t and they don’t impact real life. So, they’re not going to impact your answer. It’s just a tool that you’re going to use while you solve the problem. The conclusion there is that it’s up to you to define your axis direction, what positive direction’s going to mean, and where your 0 is.
The key is that you have to be, then, consistent with those definitions as you work the problem. If everybody defines their own unique direction, perhaps some choose positive x to the right, some choose positive x to the left, as long as everybody consistently works with their definition, you’ll see that everyone will still get the same real life physical answer in terms of where does that object end up. Is it physically here, or here, at the end of the problem? Or, how fast is it going right at the end? You’ll see this more as we go along.
We’re doing one dimensional motion. That could be either horizontally, or perhaps we’re watching a ball that we throw straight up in the air, or we’re dropping rocks from the top of the building or something like that. In which case, the convention is to call it the y-axis, to describe up/down motion.