 Hey everyone, welcome to Tutor Terrific. In this video we're going to look at the second lesson of our second unit, which covers one-dimensional motion in my physics course. I'm going to define for you speed, velocity, acceleration, and we're going to look at graphs of those three new quantities with time. So let's begin. In order to transition from my first lesson to my second lesson, we've got to think about moving from distance to speed. We all know that objects move and the initial point and the final point at which they exist is not the end of the story because we've got to look at how fast they move or what their rate is of motion. So that's how we're going to continue. We're going to also look at if that motion rate changes over the course of the trajectory. And I want to define what I just said trajectory is in the most basic terms the illustration of a path taken by an object during the course of its entire motion. So here's an example of a trajectory. Right here I have thrown, let's say, a ball off of a cliff at a certain angle and you could see the ball at certain instances in time as it goes through its path of motion. You can see it's a definite upside-down parabola. We're going to look at those in chapter three. Over here I have an electron or some negative charge that is moving at a constant velocity as far as magnitude is concerned throughout its entire motion and it makes a circle. So the trajectory is the black circle. We're going to look at this magnetic field, uniform magnetic field type motion of charged particles way later in chapter 12. So first I want to define speed. Speed is a term you've used quite a bit when you've discussed your math-word problems or your early science-word problems. And we're going to define that when it comes to physics right now. Generally speaking speed is the distance traveled divided by the time it took to travel that distance. So we have to say distance over time. You have experience with this formula when you look back at this one. Maybe in your algebra one or math one-word problems you used rate equals distance over time. There's units for speed right here. As I said earlier, I'm going to define each new quantities unit. The unit for speed is meters per second specifically. And speed does not have direction. It is a scalar. It cannot be negative. So if you tell me my speed is 35 miles per hour, I'll be fine. Even if you're not using the SI units for speed. But if you say my speed is negative 35 miles per hour, I'm going to have a problem with that because speed cannot be negative. It is a scalar. Now every time I define a quantity in this lesson, I'm going to split it into two categories. Two related parts. Average value and instantaneous value. So now we're going to look at the average speed. As we know, the speed during a trajectory or some event can change. And the average speed would be the entire distance divided by the entire time. So it completely ignores all of those changes. Because it's just looking at the beginning and end. So the entire distance traveled divided by the entire time. Now the reason there's this giant line over speed is because that line stands for average when it's put over some symbol for quantity. Now instantaneous speed is different. Instantaneous means the rate of distance traveled. The rate of change, rather, of distance in an infinitesimal time interval. So when we look at instantaneous speed, we have to take a limit. Now limit is an operation that says you allow a change in a variable to approach zero. So we're going to allow t delta t to approach zero in this expression which we see for average speed. Now what that means is we're going to look at a very small instance in time. And we're going to look at how the distance changed over that instant in time. You might say, well, it wouldn't change at all if it goes to zero. That's true. We get really small. But what does this fraction approach as we allow this delta t to get smaller? That's the idea. So we look at a very specific spot and we look at a very small instant in time. And we look at how the distance changes. Both these quantities will get really small, both the numerator and the denominator. And they usually approach some finite value, which will be positive in this case. Now you have a lot of experience with instantaneous speed when you look at your speedometer on your car. That is measuring at every instant based on the mechanics of how it's generated with the transmission, a specific speed in miles per hour, usually, or kilometers per hour if you're in the great Europe. But for our purposes, we're going to look at meters per second. Okay, velocity. Velocity. You could tell things are a little different. We usually use those terms interchangeably. But in physics, you cannot do that. These are two different terms. Velocity, generally speaking, is the change in position divided by time, not distance divided by time, change in position, which is also known as the displacement divided by time. This gives us a very different quantity. We're only looking at displacement. The units of this, however, are the same as those of speed, meters per second. We're looking at SI units. This is not a scalar. Velocity is a vector. Now velocity has a direction and a magnitude, and it can be negative. So you've got to keep that in mind. Velocity is not the same thing as speed. Average velocity is something we don't usually compute unless we're in the early stages of learning to use this quantity. We divide the entire displacement by the entire time. Now let me show you this little bar here. Again, average value of V velocity, which we just defined above. X, remember, stands for position. So some position, at some instant, minus x with this little zero. This little subscript zero is pronounced not. We get that from the good old UK. So x minus x not. What does this x not mean? It's the initial position. So some position at a later time minus the initial position divided by the total time or the change in time between those two positions. That's average velocity. Another way to write that, more simply, is just delta x over delta t. Change in position or displacement divided by change in time. So that's the average velocity. Now we know that velocities can change, and we'll learn about how to measure that in the next slide. But for now, what we usually look at when we're looking at velocity vectors, specifically, like visually, is an instantaneous velocity. What's my velocity right now? And then right now. And then right now. That's instantaneous. So we take this expression above. We take the limit of it as delta t approaches zero. So we look at a tiny little instant in time and the little bit of displacement that occurred during that tiny instant in time. This is a little bit, this limit stuff, is a little bit of an intro into calculus. So if you don't understand it, you're not going to have to compute them. I'm just defining it for all of you, all of those of you in math three or pre-calculus looking to go on in math. This is where calculus begins. All right, now, like I said, velocities can change. So we're going to look at how that's measured by looking at acceleration. Acceleration is generally defined as the change in velocity divided by the change in time. So in a certain instance of time or an amount of time, there's a change in the velocity. Well, we can measure the acceleration. And the units of acceleration are meters per second squared. So imagine how that came about. Change in velocity, that would be meters per second, because it's velocity divided by time seconds. So meters per second over seconds would mean you'd multiply those two seconds together and you'd get seconds squared. So acceleration is in units of meters per second squared in SI language. As you can tell, acceleration is a vector. Acceleration has a direction and a magnitude and can be negative. Now, the average value of acceleration would be something that's computed early on when you were learning this. The entire velocity change divided by the entire time. So we've got a certain amount of time. Velocity is at some value later on. And it's at some initial value, v0. So v minus v0 divided by delta t would give us the average value of a, the acceleration. So we use a for acceleration. And this can be more simply written as delta v over delta t. Change in velocity over change in time. Instantaneously, we're looking at the acceleration at a certain instant in time. It's the rate of change of velocity in an infinitesimal time interval. Like we did for instantaneous velocity, instantaneous acceleration is defined the same way. Take the expression for the average acceleration and take the limit of that as t, delta t approaches zero. So again, we're looking at a specific instant in time, just like when we look at a visual acceleration vector. This is an instantaneous acceleration value at this particular instant that this picture was taken. That's how to understand those. Now, let's look at graphs. We're going to be studying graphs of these three quantities with respect to time. So consider a rocket as it is thrusting upwards. The acceleration is upward and constant. Let's determine that, okay? If we introduce time into the acceleration, we will have to give it its own axis. Because when I just plot the position or something else of this rocket, it's different at different instances. There's a way to graph that so that we can see the values of these quantities at each instant in time. And this is how it's done. This isn't two dimensions, guys. It's just one axis is used for the quantity being measured, whether that be position x, velocity v, or acceleration a. And the horizontal axis is the time. So each instant in time, you have a specific position or you have a specific velocity. And you can graph those, okay? So you have some curve or some sort or a line that works and lives inside this two-dimensional space, okay? But realize that the dimension that we are studying, the way we look at our one dimensional world is now the vertical axis. And time is the horizontal axis, okay? So we have to specify that. And that's done with these little tiny symbols. So the horizontal axis will make sure there's a t somewhere. And on the vertical axis, we'll make sure there's a x, v, or an a, or all three. We'll see. So I'm going to generate one of these for you. We're going to start with some position with time. Let me give you an example. Okay. So we have some object here. And the black graph that I have here is the position line or curve for that graph. At time t equals zero, the position is also zero, okay? If we move forward in time, so let's say this is one second here. If you go up to the graph, you see the y value for that point is also one. So for example, one meter. So what we have here is that as this object moves through time, it's changing its position. It's increasing its position. For example, at four seconds, it looks like it's about at four meters. So this object is clearly moving with time. And I'm logging its position at each instant in time to get this graph. So that's, we're on to something here, but I want to generate the velocity graph. To create the velocity graph, given this graph of position, I can do that without knowing or measuring the velocity directly, okay? Again, this is a little bit into calculus, but it's something you guys can understand. What we will do to generate the velocity graph is look at the slope of the position graph. Now, it's constant because it's aligned, and it appears to be about one because as I move up one, I move over one. Remember from your early math classes like algebra one or math one, we measure slope by doing rise over run. So if you rise a certain amount and you go over and you find your particular point or you intersect the graph, you look down and look at how much you went over and that's your run. So this looks about the rise over one would be about one over one in this case. So the slope of this position graph is one. This slope gives you your values of your velocity graph everywhere, and it looks like the slope is constantly one. So the values of the velocity curve are constantly equal to that. So one, okay? Here's my red velocity graph. Notice how its values match the slope of the position graph, and that's how you generate a velocity graph with time. The values everywhere for this velocity are one, just like the slope everywhere of the position graph is one. Now to generate the acceleration graph, we would do the same thing that we just did with the velocity graph. So here, the velocity graph slope is what we'd look at, and as you can tell, the slope is flat or zero. So that means the values of the acceleration graph are all zero everywhere. So it's right on the t-axis we would say, right on the horizontal axis. So what we're doing to go from one graph to the other down this list is we look at the values of slope of the one we're studying, and those slope values become the y values or the vertical values of the next graph or curve. So the actual y values in my exercise do not matter specifically. I could actually move my position graph up or down. We never looked at the y values of the position, we only looked at the slope. So I could move this slope curve like I just did, I mean position curve like I just did, and I still get the same velocity and acceleration graphs. So let's practice generating these graphs for a given position function. For example, this one, let's say this could be an example of an object that starts far away from the reference point, moves closer to it at a constant rate, that's because it's a linear graph, the slope is constant, and then it stops about one meter away, let's say. And for the rest of time, it's just sitting at that spot. So it gets closer at its constant rate, stops one meter away, and stays there. Okay, so we have two different sections with two different slopes. So if we want to generate the velocity curve, we need to look at the initial line's slope. I've got two distinct points here, one at zero seconds and one at maybe, let's say the second tick mark is two seconds. We would just measure the slope to get from the first point to the second point. It looks like we have to go down four units on the position function's axis, we have to go over two units. So that's negative four over two for our rise of a run, which equals negative two. So if I generate the velocity graph, it has constant negative two values, okay, constant negative two values, which match the slope of the position graph in that region. Notice how I didn't continue the velocity graph past two seconds because its slope changes. Okay, the slope of the position graph is now in this region flat, it's zero. So my velocity values from two seconds onward must be equal to zero, like this, which puts it right on my t-axis, zero. Now to go from velocity to acceleration, what we have to do is we have to look at the slope of velocity everywhere. Now the slope of velocity everywhere is flat. Now I know right here we've got this problem because it's not continuous here, but we can ignore that point for this course. We don't have to draw a hole there or something saying there's a problem with the continuity. We can ignore those. As you can see the slope is zero and then it's zero again. So everywhere at zero and the acceleration graph will everywhere be equal in value to zero like this. You can see it's also right on the t-axis. So what we notice, whenever we have from this example in the previous one, linear sections or an entire linear position graph, so it has no curves in it, the acceleration will be zero. The velocity will be constant during those regions. Let's say we have this scenario. This could be an object that starts at the origin or the reference point and maybe accelerates outward. So as it's moving farther away from me, it's moving at a faster and faster rate. Now I know when I was generating this that I had to put a line here because I couldn't make the curve continue, but assume that's a continuous curve. So this is like a parabolic shape. You are going to be allowed in my introductory course to assume that when I have these curves, I'm just discussing parabolas and no other types of curves for position. So if it's a curved position graph, it's a parabola. Now when you have a parabola for your position, the velocity, or if you're looking at the slopes, are going to be quite hard to look at when it comes to instantaneous values. I see that the velocity, rather the slope of this position at the beginning is near zero because it's flat, but as we continue, the slope increases. The slope is getting larger and larger as we go outward in time. So that means my velocity values have to increase just like that slope increases, but we will assume for the simplicity that that increase is linear. So maybe the slope here is half and now it's one and now it's one and a half. Now it's two. Now it's two and a half. Now it's three. See, that's a linear increase. So my velocity graph would look like this, a line. It's showing the same values for its y values as the slopes of the previous graph. So as the slope increases, the values of the velocity increase the same way. Now to go to acceleration, I have to look at the slope, which is now constant, of the velocity graph. As you can see, it takes about two seconds over to go up one meter, let's say, or meter per second if we're discussing velocity. And so the slope would be one half. One rise over two runs. So it's one half. The values of velocity everywhere will be equal to one half like this. And notice how I'm stopping these graphs all at the same spot because there's no information past what is this six seconds on my graph. So this is how you generate these graphs and it's important to know how to do that because it helps us connect these three quantities. They are all connected by looking at these slopes. And if you have any experience in calculus, you know what we're doing is sketching derivatives. But for all of you who are not in calculus or don't plan to, you can ignore that. But they are, I do want you to know, related. All right guys, thank you so much for watching. Stay tuned for the next lesson, lesson three in this unit. And for now, this is Falconator signing out.