https://youtu.be/Z8TCIUcmL6o
PHYS 1101: Lecture Six, Part Four
Now, for the new material. I think the best introduction to this graphical analysis is just to show you what the graphs look like for the three basic types of motion we might observe.
For an object stationary, it’s moving along at a constant velocity, or it’s accelerating at a constant amount. It’s uniformly speeding up. The graphs we’re always going to be looking at here are always these quantities that characterize the motion, like position, velocity, and acceleration versus time. It’s always how these quantities are changing with time.
Remember, these are the scalar vectors, the scalar components so the sign on these graphs will mean direction, and let’s go through them and take them one at a time. So on the first square here, the first column, I’ve sketched the motion diagram that would correspond to these graphs.
So for an object that’s stationary as time goes on, it’s just sitting there. Its sum in this scenario, positive position, some place to the right of the origin. That would mean on my position versus time, for every time when I pick that time I have to go up until I hit the curve and then go over to read what’s on this axis to determine what the value of x is. And so I see for any value of time the curve is flat, so the value of x stays the same. This corresponds to something that’s stationary.
What would velocity versus time look like? Motion diagram, I know it’s stationary. Its position isn’t changing. Remember, velocity is a final position minus initial over a time interval, and that’s 0, its final initial position is 0. Position doesn’t change. That minus that gets me 0 the whole time.
Acceleration, remember? That tells me how velocity changes during a time interval. And if the velocity is 0 and stays 0, 0 minus 0 is always 0, my acceleration’s 0, too.
Constant velocity, we know the motion diagram for that. This is one representative of such a motion diagram. This is a nice, generic object moving in the positive direction to the right. What’s its position coordinate doing? As time goes on, I know that the position value increases. As time goes on, which I know are subsequent dots on my motion diagram, the position that that object is at, it increases. The object is moving to the right, and that’s what this graph says. As time goes on, my x-coordinate, compared to the origin, gets larger and larger, and it’s positive because it’s to the right of the origin.
Given v is the change in x divided by the change in time, this tells me two things. On one hand, Δx over Δt is a slope of this line. I also can just think about my motion diagram that this corresponds to and know that the velocity is constant. The object is moving to the right in a straight line, constant speed. Velocity is some positive value. At all times it’s the same value.
Acceleration, again, being a representation of how the velocity is changing. The velocity is not changing, 2 meters per second minus 2 meters per second is 0. Acceleration is 0 for this.
Constant acceleration. If you think about what the x plot versus t looks like, let me just point out that these are all equal time intervals, say, equal time intervals, and at these equal time intervals, I know that my position is moving to the right, but I get further and further between each time interval. Between each time interval, think of this graph. Let me sketch here for you a couple of x values. Let me do three, and here’s at another time. I see that as these equal time intervals tick by, I actually get further and further, and that leads to this curve in that plot, x versus t.
What’s the velocity in this case? If this is a nice constant acceleration, I know that this arrow represents the same Δv that I’m adding every time to stretch that velocity vector. So that would mean that between all these time intervals my velocity is increasing by the same amount. The next time interval is a straight line. Again, the velocity increases by that same amount. That Δv there represents our acceleration.
So if something is speeding up, its position keeps increasing more and more, its velocity increases now at a steady rate. These arrows get longer at a steady rate, and the acceleration is the amount of this increase every time. This is my Δv. That’s constant. It’s some constant, positive amount here. So for all these time intervals, each one of these time intervals, the Δv is the same amount. It’s the same positive, constant amount.
Okay. With that summary, we’re going to do some examples now of graphs. I think the best way to get familiar with them is just to practice. So I have a lot of quiz questions here for us, and I’m going to help you through quite a few of them to get you warmed up to this idea.
Here’s some bullets here for you of suggestions to think about as you’re faced with a graph, and you’re asked a question about it.
First thing, often people rush into a plot, and they just assume they know what they’re looking at, that it’s a plot of position versus time, when actually it’s showing you how the velocity changes with time. So slow down. Step one, be sure you understand what it is, what information is being shown to you.
Then pick two different times, two different points on the graph, a bit like I was doing above here where I was dashing some lines. Pick two different points and just physically be sure you can picture what this means. This means at an earlier time, at this earlier time I was at a positive position but kind of close to the origin. At a later time I had steadily moved on and was now at a larger positive x-coordinate.
So picture in your mind, what does that mean? Use, at least, 2 of these points to help you picture what that motion diagram must look like. And that’s my point 3. Try to picture those two points. How do they compare? What do they physically mean? From that, can you generate in your mind or imagine a rough picture of the motion, and what then the motion diagram would look like.
Okay. Before we get on to our examples, I want to emphasize to you this notion about the slope on a curve and the usefulness of it of looking at that slope. Let’s start here and imagine we’re looking at a plot of position versus time. Let’s take a scenario where the position is just smoothly increasing so I have a straight line here.
I remind you that the definition of velocity is a change in position divided by a change in time. When you’re looking at a plot of x versus t, you realize that that change in x is a rise. Here is an x final, an x0. So x minus x0 is the rise. And the t minus t0, that’s a run. Here is a t0 and here is a t so that the interval there, the length of that is t minus t0. That’s run.
So these terms, rise over run, you probably remember from a class long ago as being the slope of that line. If you do this difference carefully and properly, it’s always final minus initial, final minus initial. The sign of that slope has meaning, too. If there’s an upward slope to that line. those quantities will be positive. If you ever have a line that slopes downhill, the slope would be negative. You’d end up with a negative quantity of x final minus x initial.
Okay. The rise over the run on a position versus time plot is the velocity. The motion is just steadily increasing. I’ve got a straight line here. As you saw above, that means a constant velocity. The slope of this line is the same. Whatever you calculate over this interval, that’s the same slope anywhere.
Just to remind you of that, if you have a straight line on a plot, the slope is the same everywhere for that straight line. So you can take any two spots and calculate the rise over the run to figure out the slope. That number will be the same for any intervals that you pick along a straight line.
The next level is to think about a velocity versus time plot. The same idea applies. Remember that acceleration is v minus v0 divided by t minus t0, and on this plot, velocity versus time, the rise, the v final, and the v initial, that’s our numerator here for acceleration, v minus v0. And our run during that time interval, t minus t0. Let me go ahead and write that on here again, this is our run, and this is our rise.
And so, again, the ratio of rise over run on this plot tells us useful information. It is giving us the value of acceleration, the slope. The slope of velocity versus time curve tells you about the acceleration.
If you think about it, the units make sense, too. The rise over run on a position graph has units of meters per second, which is velocity. On a velocity versus time graph, the rise will have units of meters per second, and the run will have units of seconds.
Let me write that to the side here, meters per second. That’s units of acceleration. Let me write it over here again for us. It has units of meters per second. Okay?
So we’re going to be looking at some of these graphs below. You’re going to have to be calculating slopes to answer some of the questions, and you might want to go back to this section to just remind yourself what it is.
So to get a real number, for example, let’s see what’s the slope of this line. The rise is going to be x of +12 minus x0, which is 4, 12 minus 4 would be 8 meters. And then what would the run be over that same interval? I saw a change of 8 meters. That was over a time interval of 2 seconds. So the velocity for this object is 4 meters per second.