 Hey everyone, welcome to Tutor Terrific. Today we're going to look at my third lesson in my second unit of physics. We're looking at one-dimensional motion in this unit and specifically in this lesson we're going to look more at some more graphing examples like we did last time. We're going to do some simple calculations involving speed velocity and acceleration and then I'm introduced to you the big three. These are the big three equations of kinematics for constant acceleration. We're going to look at those for the next few lessons to be exact. They are going to play a role in so many things that you do in kinematics and dynamics. It's not even funny. So let's get started. All right, we're going to do some more graphing to get started because I gave you some examples last time but I believe you need some more. We're going to continue to create position velocity and acceleration graphs but I'm not just going to give you the position graphs to start. You are going to be given some scenarios and be asked to make the position graphs as well from those scenarios. Then after that we'll use the methods from last time to generate the velocity and acceleration graphs. So let's say you're on an elevator and it's on the eighth floor and it descends to the second floor in eight seconds. How can we use this grid to denote that information? Well, we know that the vertical graph is our position. It's the one-dimensional world we live in and so we can treat each of these tick marks as a floor if we wanted. So we're not in SI units per say on the vertical axis but we're still using it. In eight seconds we could use each tick mark on the horizontal axis for time as a second. One, two, three, four, five, six, seven, eight seconds. So from zero to eight seconds you have to go from the eighth floor to the second floor and it says at a constant velocity. So that means it's moving down at a constant rate so it should be linear. So let's see what we generate here. The eighth tick mark down to the second tick mark from zero to eight seconds constant rate of decrease or descent. Perfect. So that's our position function with time in terms of floors. Now how do we get to the velocity graph? Well, we need to look at the slope of the position graph. Now if we're careful here we're going to see that we run eight but we rise one, two, three, four, five, six negatively. So that's a negative six rise over an eight run. Negative six eighths is equivalent to negative three fourths. So that's the slope of this graph simplified. Negative three fourths and that's a constant slope. So that slope becomes the values of the velocity graph. So they have one value everywhere. Negative three fourths. We should have a constant line at negative three fourths on this graph for velocity and that's exactly what I generated here. Now to go to acceleration we look at the slope of velocity. We could see that the slope is constantly flat. It's zero. So we get zero value for acceleration right on the t-axis. So I want you to keep a note here whenever your position function is made up of linear portions or is completely linear the acceleration will be zero. All right, on to another example. We have a cannonball and it's dropped from the top of a cliff and it hits the ground sometime later. So if we're dropping something it starts out by having no velocity and the farther it goes the more velocity it gets because it's under acceleration. We're going to look more at that in a little bit later lesson but this is an acceleration that we're under. So we would expect with time that the slope of the position graph would decrease meaning it would get more negative. We'd expect this thing to start falling really slowly. Basically start falling with no initial speed or velocity and then its velocity would decrease becoming more negative with time. So this kind of graph would result from that. Something like this. It actually looks like it's being thrown outward but remember we are only looking at one dimension of position. In this case that would have to be the vertical position because we cannot look at the horizontal position. It wouldn't change if it's being dropped straight down. So this is a graph of vertical position. So our one-dimensional position number line is vertical in this case and as we increase time we can see that it falls faster and faster to the ground. So that's our position function. It's a curve and I told you last time when I give you a curve position function you assume it's parabola meaning that the velocity will be a line. Now the velocity starts out. Let's look at the slope. The slope of the position function at first is 0. It's flat and the slope decreases and becomes more and more negative. The velocity function the graph should start out at 0 and get more and more negative in a linear fashion like this. Perfect. So the slopes of the position function match estimatively because we don't have exact values. The velocity functions values. Slopes turn into values for the next graph. So the velocity is linear and it's negative sloped. It's some value which we could look at. It looks like about we run 3 over before we rise 1 somewhere between there. So the acceleration's value is something negative and it's some negative fraction less than 1. Like so. About negative one-half it seems. So that's the acceleration graph. You can see it's constant. These are constant accelerations. So that's another example of making graphs. Now I want to visualize some of these vectors in two simple scenarios here. Let's say a car accelerated to the right. These units are not SI units, kilometers per hour per second, but it does help us illustrate how the numbers match. I am sorry for the glare here. So at the start the car has 0 velocity at time t equals 0. According to this acceleration, every second that passes it will be going 15 kilometers per hour faster. That's what is meant by 15 kilometers per hour per second. So in the first second we're going 15 kilometers an hour. The next second, 2.20 seconds, we've added another 15 kilometers an hour to its velocity. So now we're going 30 kilometers an hour. Now if we look 5 seconds ahead we've added 15 kilometers an hour 5 times. That's 75 kilometers an hour. As you can see the velocity vector, this little green vector here, is growing in size because we can relate all these because the same vector is the same unit. Now what if the acceleration was pointing in the opposite direction? I'll have that for you over here. Let's say the car is moving to the left and it was moving at a certain velocity and with time that velocity vector would decrease in size since it's moving to the left because the acceleration points in the opposite direction, the velocity vector will decrease. Eventually if the acceleration vector continues to act the car will actually turn around and start backing up so its velocity vector points in the same direction as the acceleration. So these are two examples of acceleration in the same direction as velocity, acceleration in the opposite direction of velocity. We often call this scenario over here the second one deceleration, but it doesn't have to be called that. Alright now I want to do some examples with you so we can use the equations for the speed, velocity, acceleration that we learned last time. We can use those in this lesson because we need to get used to computing in physics doing some calculations and so this is basically our first start at doing that after estimation in unit one. So in this problem we've got a position of a runner as a function of time and it's plotted as moving along the x-axis of this coordinate system here doing a three second time interval. Notice how it was given three significant figures, two significant zeros, a decimal and a three and the runner's position changes from x1 so its initial position is 50.0 meters and its final position x2 is 30.5 meters and that's shown in this figure right here from 50 to just bigger than 30 meters positions. What is the runner's average velocity? That's what we're asked for. I want you to notice what this vector is, this blue vector here. Notice how it's called delta x. We've got a start and finish position so this is the displacement vector. Now what we're going to calculate though is the average velocity. Now if you remember from last time average velocity is equal to final position minus initial position over the change in time or delta x over delta t. We need to compute delta x from the two particular positions so we would do final 30.5 minus initial 50. We're going to get some negative value for that just under negative 20, negative 19.5 meters so that's its displacement. We have to keep that negative sign here because displacement is a vector. Now we will divide that by the time which was told to us three seconds and so we got negative 19.5 divided by 3.00 and that would be negative 6.50 meters per second. Notice how I've still preserved the negative sign. Velocity has a direction and in one dimension it's either to the left or to the right or up or down. Those are our only choices and they are dichotomous pairs so they only work with each other. You can't mix them in one dimensional motion. So its average velocity was negative 6.5 meters per second. Notice how I didn't ask for instantaneous velocity so I don't know how fast it was going at every spot but if we assume it was constant then this is its velocity everywhere. Alright, next problem. This is going to ask us to compute average acceleration for the following example. We've got a car accelerating along a straight road from rest to 75 kilometers per hour and it took five seconds to do that. What is the magnitude of its average acceleration? You might recognize this from the previous couple slides where I showed you examples of acceleration and velocity vectors. It's the same situation. This problem wants specifically the magnitude of this average acceleration. So this is the equation we're going to use. Average acceleration is final velocity minus initial velocity over the change in time or just delta V over delta T. Okay, but we've got a problem. The 75 kilometers per hour that is not the SI units for our velocity. We need to turn this into meters per second. So we have to convert the kilometers to meters and the hours to seconds. So this is done with the factor label method that I showed you. This is a compound unit conversion. I'm going to start with the kilometers. I'm going to convert that to meters by multiplying this number by 1000 meters for every kilometer. So 1000 meters over one kilometer. That's a prefix unit conversion. The kilometers will cancel. Then we will take care of the hours. I want seconds on bottom. So I will put a conversion factor from seconds to hours. So it's one hour is equal to 3600 seconds. I'm multiplying that on next. So the hours are going to cancel here. We're going to end up taking 75 and multiplying it by 1000 and dividing it by 3600, which gives us approximately 21 meters per second. Now the initial velocity wasn't given to us except in coded language. This word from rest. Rest means zero meters per second. It means we are not moving initially. So the initial velocity that we'll plug in here for v0 is zero meters per second. Plugging everything in, we have the five seconds for delta T. We have the 21 meters per second for the final velocity in the correct units and zero meters per second for the initial velocity. Now when we do this calculation, we get about 4.2 meters per second per second. As you know, acceleration, this is what occurs and those two seconds multiply together to get meters per second squared. 4.2 is probably not as accurate as I could have been with the calculator's raw results. However, I only had two sig figs to start with for both of my initial measurements. And so we're doing some dividing. We have to look at the number of sig figs and that determines the number of our final sig figs. Both the initial ones have two. So my final one can only have two sig figs. Alright, last example. We're going to have an automobile that is moving to the right along a straight highway. And we choose the positive x-axis as the number line. So that's our number line situation. It's a horizontal number line along the straight highway. The driver decides to put on the brakes. Now, if the initial velocity when he hits the brakes is 15 meters per second and it takes him five seconds to slow down to 5.0 meters per second, what is the car's average acceleration? And it looks like we have everything we need. However, you have to realize that we're slowing down. So we might have a peculiar situation here where the acceleration vector points in the opposite direction as velocity. But we're going to start by using this equation. And this time we have no conversions to do. Everything is in the proper SI units for us. So we have the final velocity 5 meters per second minus the initial velocity 15 meters per second divided by the time 5 seconds. And that gives us a negative number for our answer. Negative 2.0 meters per second squared. Acceleration is a vector. You preserve the negatives to show that we chose the positive direction as to be in the direction it's moving. And so acceleration has to point in the negative direction because it points the other way. All right, so this situation is what we're getting at here. Acceleration is pointing in the opposite direction of velocity. All right, now I've been excited for this. The big three. Now this is literally a big three. And this is not what I'm talking about here. I'm going to show you three kinematic equations for motion in the next few slides. Okay? I want to look at how they're derived so I don't just give them to you and expect you to just memorize them. It will help you if I show you how they were derived. Okay? Now, what we must assume in these equations is that the acceleration is constant. Okay? It could be zero, but it's definitely something constant. Okay? The acceleration is not allowed to vary. We are going to be working in one dimension only. Okay? I know there, in our world, everything moves in three dimensions and you see a lot of stuff in your video games or whatever you do on your papers or even in the textbook if you looked ahead where the dimensions of motion are two dimensions, but we are only assuming one dimension of motion right now. So a number line is the world we live in. We're going to start our derivation of these equations with an equation for acceleration with respect to time. Now you might think, wait, you just said the acceleration has to be constant. That's correct. So that's our starting condition. Now if you look specifically at this equation, all we have is just this simple equation for acceleration. A of t equals some constant value of a whatever it happens to be. It's not allowed to change with time. So we're going to start with that. Now we're going to go backwards and get the equation for velocity. What we would do is we will, and I don't have to memorize this, I'm just showing you how this works, is we would take the acceleration as a constant and put a t by the variable for time and then add a new constant for velocity, initial velocity, on top of that. And so that's how we get from this acceleration to this velocity equation, v0 plus at. Okay. Now we're going to take this back one more step. We're going to get back to position with this equation. We're going to take the constant of this equation and multiply it by t and add a new constant in front of it, which will be x0, the initial position. And we're going to take the old at term and we're going to put a 1 half in front and a squared next to the t. And so we get this. Okay. It seems like this process could continue on, but there's nothing to continue on to. Position, velocity, and acceleration are three main quantities that are vector quantities in physics for motion. And so we stopped the process here. Now, to be truth be told, we've actually derived two of the three equations already. What we just did, you may not understand the rules for it, but you don't have to because what we just did requires calculus. I'm just showing you what the results are. But all of these equations involve time. Okay. What if we didn't want, or what if we couldn't use time? And we needed another way to study one dimensional motion that had acceleration, but we didn't have any time information. Well, we're going to take the two of the big three equations that we've already derived, which are the second and third ones on the other page. And we are going to do the following. We are going to substitute things into them so that time disappears. If you do that, I'm not going to show the whole thing. If you do that, you substitute out time algebraically, you get the following. The final velocity squared equals the initial velocity squared plus two times the acceleration times the displacement x minus x naught. So that's what occurs when you just take these two equations and make that happen. This x refers to x of t, and this v refers to v of t in this equation. So now we have something that is independent of time. This is called a time independent equation in kinematics. So these three equations right here are the big three, and here they are written in nice, beautiful math script, the big three kinematic equations. Notice how the first one has no position information in it. It's just the final velocity equals the initial velocity plus the acceleration times the time. The second equation is the most complicated. It is a quadratic equation in time. It's the final position, and it equals the initial position plus the initial velocity times time plus one half times the acceleration times time squared. Then the final time independent equation is the final velocity squared equals the initial velocity squared plus two times the acceleration all times the final position minus the initial position. In other words, the displacement in this parenthesis. These are the big three kinematic equations. We're going to spend the next few lessons learning them. Now, if the acceleration is zero, we're going to get a different version of the second equation, which we would get a different version of all the equations, but they would be meaningless. If you look at the first and third equation, if you plug in zero for a, you just get that the final velocity is equal to the initial velocity. You would expect that if it's not accelerating because it would have a constant velocity. The second equation is actually unique if a is zero because you still have position changes. The final position would be equal to the initial position plus the initial velocity times the time. That's an extra equation. This is like a little wart on the big three. It's nothing new. It's just one that's used all the time because there's lots of times when a is zero. Here they are, guys, the big three kinematic equations. You will not have to memorize these. You'll always have access to them in physics, but it's good to become familiar with them. That can only be done by doing problems. We're going to look at that next time. Guys, thanks for watching this video. Lesson three of unit two. Stick around for the next one. It's coming soon. This is Falconator, signing out.