https://youtu.be/s-LHaaD7frk
Now that we have this curved motion, this two-dimensional motion, we’re going to find that in order to do the mathematics, all of the basic vectors that we’ll need are going to have to be broken up into their x and their y components in order to mathematically solve the problem. Or doing this makes the mathematics much easier to solve.
What are our main vectors? Very similar to 1D, we’re going to have a vector that describes position, velocity and acceleration. But why do we have to do this? I want to give you a little scenario to walk through to motivate why this is useful or helpful. Do this quick demonstration on your own as I describe it. Presumably, there’s a glass of something sitting in front of you as you watch this movie.
So, a lineup, imagine the table here is defining the edge of the table, a horizontal direction. Here sits your cup. Push on your cup, so you move your glass away from you and at an angle. So, you move your cup so it slides at about 45 degrees for some distance.
What I want you to appreciate now is that you can end up with the exact same motion if you push on your cup instead of directly in the direction you want it to go. But you push straight toward it and also directly to the left. And by balancing, you’ll feel it, a combination of these two. You can move the cup in the exact same direction off at an angle.
The point is that the result, the real-life physics of the motion that you’re causing; it’s the same whether you use a single push at the angle or you use a combination of these two smaller pushes at an angle.
This equivalent to this on one hand seems more complicated because I’ve introduced two smaller pushes. The beauty of it, though, is that by breaking it down like this or realizing that this equivalent to this, I have turned the problem into two one-dimensional problems; vertical and horizontal. The combination results in this motion off at an angle.
But because I only have vertical vector components to deal with and I’m going to have only horizontal components to deal with, it’s nice because the mathematics, then, I can handle by just using a sine to indicate direction of vectors again. And that’s what’s going to make the mathematics easy for us.
The key is going to be that while I treat all of the horizontal vectors and all of the vertical vectors separately, the time variable that impacts both the horizontal and the vertical is going to be the same time variable. Because these two things are happening at the same time. You had to push at the same time with both of these to get this cup to move 45 degrees.
So, let me show you, motivate it in a slightly different way by showing you a little movie of just the motion of some object. And you can really see how motion that can be off at an angle is really just made up of a vertical part and a horizontal part.
Okay, I’m going to let you listen here to the audio here, as well.
Video: This toy bulldozer, which runs at a constant speed, will be used to show how velocities add together in two dimensions. The bulldozer is set on a paper sheet that can be pulled along the table at a constant velocity. The bulldozer is first set up to run directly across this sheet and it follows this path.
This time, the sheet will be pulled to the right at half the speed of the bulldozer. What will the path of the bulldozer be?
Lecturer: Okay, having watched that, let me add some sketches here to emphasize what it is that you were seeing. The first scenario that they showed you where the bulldozer was moving just along its straight path, it was following this straight, let’s call it y-axis motion and moving along with a certain velocity.
So, if I were to go in with a black pen and do a motion diagram of that, let me add about four points or so just to convey the gist of it. And then add some velocity arrows here, each of which, of course, is the same length. That would represent this constant motion and that straight path that that object followed.
Let’s go sketch, quickly, a top view of this motion, as though we were looking down on this blue sheet. This is my y-axis, this is my x. And here is the motion of my bulldozer moving just in the y-direction.
Now, in the next movie, the direct path that that bulldozer took followed this path along at an angle. We know that on the paper, it still experienced the same velocity moving straight along the paper. But the paper was being moved to the right at a different velocity. So, here, this velocity component; that of the bulldozer, I would call a v-y. And at the same time, then, the blue sheet was being [inaudible at 00:07:00] along in the x direction with a certain velocity that I’m going to call the “x.”
The motion that resulted, the real path that you saw this object take, was the combination of these two. It really is the vector’s sum of the v-x and the v-y at each instant. Let me draw the real velocity vectors. In fact, let me draw them with a thicker pen to emphasize that is really what our eyes saw as the motion for this object.
If you were to do the top view, again, of it, it makes parts of it more clear. I end up with, while I have the same y motion, I have these dots for my x motion. Leading to really, what’s the vector’s sum as my total motion for this object? Its constant velocity, meaning it’s equal length of all these vectors. But the real motion is along this angle.
Here, I have my v-y components. And here I have my v-x components. And it’s the sum, v-x plus v-y. V-x plus v-y, it gets me my total velocity, my hypotenuse for that motion. So, we’re going to take this motion that’s off at an angle that’s going to potentially follow an arc. And we’re going to break it up into only the vertical parts and the horizontal parts.
And that’s because the combination does give us the right physical behavior of this object. But by only looking at horizontal separate from vertical, we’re going to be able to use sines with our numbers, so that our mathematics goes smoothly. So, you’ll see more what this is as we go along.
Here’s my steps, though, so here’s what you’ve got to do. Two-dimensional motion where we have an arc. All of the vectors, we have to separate into the components. And this is because it’s the x components that position velocity, acceleration, the x parts to those only influence each other.
Similarly, the vertical components of these key vectors are only going to influence each other. And so, we need to separate out and handle these two parts independently. So, here is my plot of position, velocity and acceleration; three examples of key vectors. Let me get a large pen and emphasize what we’re going to do.
The real visual vector that we are going to see or observe in the motion is hypotenuse. For position, as an example, that object is sitting away from the origin off at some angle. But we’re going to describe that, we’re going to break that position up into its x-coordinate and its y-coordinate.
So, the way we mathematically write that is that this position vector r is made up of two vector parts. It’s the x scalar component times the x unit vector plus the y scalar component times the y scalar vector. X and y are the coordinates of that position relative to the origin.
In the same vein, our velocity vector, the hypotenuse is going to represent the real direction of motion, real speed that that object is headed at at one instant. But we’re going to work with the vertical part of that and the horizontal part of that.
This means these two vectors here, I would call v-x and the v-y. And I mathematically write that my total velocity, my speed and the real direction the object is moving, that that vector is equivalent to the horizontal scalar component, times the unit vector to indicate x direction, plus the y scalar component for that velocity times its unit vector.
Acceleration, likewise, now that we’re in two dimensions, can have a horizontal and a vertical component. It can be off at some angle, that total acceleration. So, we would write that a is made up of an x vector and a y vector. Or that the total acceleration vector, the hypotenuse, is equal to the x scalar component times that unit vector in the x direction. Plus [inaudible at 00:13:01] times the y unit vector.
So, let me emphasize with a highlighter that the scalar component for position, scalar components for velocity and scalar components for acceleration are going to be our mathematical focus. Our real life, tying it to the problem focus will be the hypotenuse. And we need to go back and forth between these two descriptions. The components versus the hypotenuse.
What’s highlighted in yellow here, these scalar components, are what we’re going to use to do the math. So, let me highlight that, that’s what I’ve got highlighted there in yellow.
So, here’s a quiz question for you. To appreciate this going back and forth between the real-life description, which would be the real direction a ball is headed and its speed and then this component description. So, work this problem out.
As viewed from the side, a soccer ball is kicked to the right at an angle of 30 degrees from the horizontal. It has a speed of 23.2 meters per second. That describes a velocity vector, an initial velocity vector. Sketch that vector. Then, with the standard axis definitions, determine what the x component is of that vector. So, now, you’ve got to go to your trigonometry to figure that out.