https://youtu.be/jiuilSeQn8k
PHYS 1101: Lecture Five, Part Two
Continuing with our warm up, here’s my summary for you of what we covered the last lecture. It was a big lecture. It had some important components that we have to be sure we really understand well before we can move on.
Last lecture the focus was on defining and understanding the specific and exact variables that we need, terms that we need, to describe uniquely the location of an object at an instant, how fast it’s going at that instant, and what direction, and then at that instant, is the object speeding up or slowing down? Do we have acceleration? For each of these I have made up a table here and I want to walk you through what the equations are, the definition of the terms that we needed to define these uniquely, units. I want to talk about the physical meaning if these quantities are constant or changing as time goes on.
As a first statement, though, before we can define any of these uniquely, before we can solve any of these problems, we have to define an axis, a positive direction, and an origin where the location of 0 is going to be defined. That’s the only way that these variables that we’ve introduce for location, how fast, and acceleration are going to have any meaning, because we have to use vectors to describe each of these, and the direction of those vectors captures important physical meaning for us.
Okay, let’s start with the equations that we use to define these. Let’s start with location. For location we defined two unique vectors, I don’t have an arrow written over these, what I’ve written here are the scalar components. These are vectors, x and Δx, that I write. The sign for these one-dimensional scalar components are going to tell us the direction for these vectors.
We needed a position vector and a displacement vector. Position would be the actual location and then the vector would be from the origin out to where that location is, say, if our object was here at an instant. The magnitude would be the little distance from the origin, a straight line, and then the sign would tell if we’re to the right or the left of the origin. Displacement is a useful variable when position is changing as time goes on. Displacement is a comparison of two position coordinates, the final minus the initial.
Let’s work our way down the column here. For location what would be the meaning if our location variable was consent? That would be, the object is standing still and it’s at some specific coordinate. It could be positive or negative depending on what side of the origin we’re on, as I explained above.
What’s the meaning if this variable is changing? Once you have motion and the position is changing a motion diagram is a great way to visualize the nature of that motion. Once I have change then what’s going to be natural and useful to work with is the displacement vector, the vector that compares where the object started to where the object ended up. If the object moved to the right, then this displacement vector will end up pointing to the right and be positive. The magnitude will be the literal length, the distance between the two spots. I could have a negative displacement vector, too, which would point in the negative direction, indicating the object moved and ended up to the left.
Okay, let’s go up to our second column, “How fast?” To capture uniquely how fast an object’s going we had two ideas to consider. In some cases it’s sufficient to talk about an average velocity. More often than not though, we’re going to need to think about what the velocity at an instant, at every instant, say, during the motion or comparing an instant initially to just at the end of a problem, the final scope.
We’re going to need these instantaneous notions, definitely, for cases where an object is speeding up or slowing down. The instant at the beginning, the initial velocity, we’re going to use the variable v0, and then for the final velocity just the variable v. Velocity always has to be a comparison of two positions and you need to know what the time interval was that resulted in this change in position. That’s what velocity tells you, how fast the thing is going, the object is going. It’s going to have units of meters divided by seconds or meters per second.
Going down the column, what’s the meaning if our velocity is constant? That would mean physically that the object is moving along in a straight line. My velocity vectors in my motion diagram all point in one direction and they all have the same length. If this is my axis’ definition then I’ll know mathematically a positive value for v represents positive direction motion. A negative value for v would mean motion going to the left. To emphasize again, any time you’re going to look at the motion at any instant, the velocity vector at that instant always have to point in the literal direction that that object is headed.
What’s the meaning if our velocity is changing? We’re focused one 1D motion here, so we’re going to restrict it to motion along a straight line. If the velocity changes, subsequent vectors either get longer or get smaller. This would be a motion diagram, again, of a positive velocity. This object happens to be speeding up, but I could have an object moving to the left that, in this example, is slowing down.
Let me point out, while I’m down here at this end of the table, that there’s an analogy here between the previous variable, the notion of location, when it changes in time it gives us the same picture as the variable to the right, velocity, when it’s constant. Notice that these two scenarios describe, it’s the same kind of motion for an object that has a steady position that’s changing with time. That means I have a constant velocity. That follows purely because of the definition of velocity. When velocity is constant I have a constant value for changing position over some time interval, so v tells us how position changes.
Let’s go on then to the acceleration, which analogously tells us how the velocity changes, a is defined to be the difference between two velocities divided by time, that time interval. The units of acceleration have to be meters per second divide by second. That’s a great way to think about it, because it adds physical intuition and meaning for you, I think. It means every second my velocity has to change by some many meters per second. You mathematically can write that the equivalent of this is meters per seconds squared. So if ever you see a quantity in a problem that has units of meters per seconds squared, that is a value for acceleration. Quantities with units of distance over time, meters per second, that’s a velocity, and of course, distance is units of meters.
In our class we’re going to restrict ourselves to studying motion during an interval for which the acceleration is constant. By that I mean that the acceleration — well, let’s go down to the next table here, let’s go down.
What would be the meaning of constant acceleration? Let’s just take this top motion diagram to emphasize it. Constant acceleration would be, as you compare subsequent velocity vectors, they stretch or get smaller by the same amount, e.g., a might be equal to plus 2 meters per second every second. a being constant means for every second during the duration of this motion this is how much I have to change the velocity vector by. So in our motion diagram ever vector has to increase by this amount to describe the next velocity vector.
Okay, I’ve got four scenarios drawn here for you. It looks a little complicated, but it’s important to see these four different scenarios. This scenario, the object moving to the right and this scenario with the object moving to the right match more of our physical intuition of what the meaning of signs should be for these quantities. That means an object moving to right with a positive axis defined to the right would give us positive velocity vectors, the object is speeding up, my acceleration has to be in the same direction as v. That vector points to the right and so it would be positive. Intuitively we really strongly want to assign a positive acceleration to when objects are speeding up. This scenario matches your intuition.
This scenario does too. Positive velocity to the right, the object is slowing down, the acceleration vector has to be opposite in sign, opposite in direction to accomplish that. Because it points in the negative direction I need to use a negative a value. That matches your intuition, negative a does seem somewhat natural to describe an object slowing down.
Okay, but because in physics we’re using this mathematical tool to handle these variables, as I’ve emphasized all along the way, the sign that we use doesn’t have to do with speeding up or slowing down, it only has to do with the direction of these vectors. That’s how we need to interpret the sign for the mathematics to work as well as it’s going to work for us. This means if you consider a scenario, this one, the second one and the fourth one, where the velocity is to the left, all of a sudden now you’re intuition of what the sign of a should be is going to be counterintuitive.
Let me walk you through these two scenarios. If the velocity is to the left I will make that a negative quantity to indicate, I want to be able to describe objects moving to the left or the right. If it’s slowing down the key is that a has to point opposite to that velocity. Consequently, it’s going to going to have to have the opposite sign, it’s going to have to be positive. Positive a will cause this object to slow down if it’s moving to the left.
Likewise, if an object is speeding up, a has to point in the same direction. That’s the only way these vectors get stretched and get larger. Negative v then means I have to have a negative a, in spite of the fact that the object’s speeding up. This negative, it’s necessary for the mathematics to give us a velocity that is increasing in the negative direction.
The last section I have here describes for you the picture of changing acceleration. You have to watch carefully for this, because the tools that we’re going to use are only valid over time intervals for which a is constant. This vector, between all spots on the motion diagram, has to be the same length. It can be 0 or it has to be the same length.
Here’s a cartoon I show you of some motion where from start to finish the acceleration is not constant. Notice during this first time interval the velocity vector is stretching always by the same amount. This looks like it represents constant a, for example, 2 meters per second every second is what my change is, but right at this instant the motion changes. The characteristics are different. Now the object’s moving along at a constant velocity, a is 0 for this part of the motion, so from start to finish my acceleration has changed. It is not, say, 2 meters per second every second the entire motion.
I can do this problem if I break it up into two parts. Instead of making the finish here, I could do this part of the motion where a is constant, with one set of variables and equations and I’ll do an example of that for you today. Likewise, I could then connect the results I have from this motion here to a second problem for which a is 0 now for this motion. At this instant, that instant to corresponds the end of the first part will correspond to the start of the next section of the motion, and I can connect some variables here at that instant, because they will represent the same thing.
As an example, the final velocity, right here, from the first part is the same as the initial velocity for the motion here, for the second part. Okay, so be on the lookout for cases where acceleration really is constant. As soon as it changes you’ll see that we’re going to have to break the problem up and to do it in pieces.
The last thing I’ll highlight for you is the same thing I emphasized here. Follows with the connection of a change in velocity is a constant acceleration. These arrows that I draw — in fact I should draw arrows on both ends — they really describe the same scenarios. That, again, is a consequence of the fact that a tells me how velocity changes. So constant a means my velocity’s changing at a nice steady pace. My arrows are just getting stretched by the same amount as time goes on.
Now it’s time for our reading quiz. This is where we warm up for the new lecture. These sections that we’re going to lecture on are just using these definitions of position, velocity, and acceleration to solve some problems and here are two quiz questions for you.
To start our new lecture material let’s do our two reading quiz questions as part of our lecture quiz, so these questions have to do with content in Sections 2.4 through 2.6. The emphasis in the questions is, again, similar. It mostly emphasizes variables and memorizing what they mean very specifically. I’ll let you pause the video here, read these questions, and answer them.