Kinematics Describing Motion in Two and Three Dimensions The Position and Velocity Vectors Page [1 of 2]

We’ve talked about motion in one dimension, moving forwards and backwards, or up and down, and if you restrict yourself to one dimension, and if your acceleration is constant, we’ve pretty much solved the problem of kinematics. If you give me initial position and velocity and acceleration, I can describe the entire motion for all time. It’s very nice and it’s really useful in a lot of problems.

But, it’s also true that the world is not one-dimensional. I can take a ball and toss it up in the air and it could go in an arc. That’s not a one-dimensional problem. It’s going both up and down and left and right. You could imagine watching a boat from above and it’s tacking its way across a lake. Its motion is intrinsically two-dimensional. The word two-dimensional just means that you can move left and right and forwards and backwards. Ultimately, we’re going to want to talk about three dimensions where you can also go left, right, forward, backward, up, down. It’s already a kind of a conceptual leap to go from 1-D to 2-D, a little bit of math to learn. Once we’ve made that leap, going from 2-D to 3-D is a snap.

So, what’s the difference between working problems in two dimensions? There’s an awful lot of Physics problems that are two-dimensional. Tossing a ball, you might think it’s three dimensional, but the ball really moves just in this plane. When I throw it, it’s not moving forwards and backwards. So, it’s physically meaningful and useful to work two-dimensional problems.

What are we going to talk about? Well, all the quantities that kinematics describes – position, displacement, velocity, and acceleration – these are all quantities, which we now need to think about in two dimensions. Example: Here’s an object, which is just sitting on the piece of paper. How am I going to describe its position? In one dimension, I lay down a coordinate system and I say it’s at zero, or let’s put it here, it’s at negative one. In one dimension, you just need a number. The number might have a sign, ¯1, but it’s just a single number. In order to come up with the number, you have to pick a coordinate system and pick a place you want to call zero. It’s the same in two dimensions. You need to pick a coordinate system and in two dimensions, your coordinate system will have both an X and a Y direction and an origin. So, where is this object located? Where is its position? Instead of just a

number, like ¯1, I’ve got a totally different coordinate system. It’s a little confusing having two, so let’s just get rid of 1-D. Let’s work in 2-D now. In 2-D, I would just draw an arrow from the origin to the object. That’s the way you describe where it is. You need a vector. And I would give that vector a name. You can name quantities whatever you want. We’ve been calling position X in 1-D, but since now we’ve got X and Y, we’d like a new name. People usually use R . So, it’s an R vector. That’s the position vector. And since this is where it initially is, I might call it iR .

So, now we can talk about motion. That’s what kinematics is about. Imagine that the little creature, whatever it is, walks from here to here, so it ends up at a new position, let me have him walk over this-away, and this is where it ends up. Its motion was a little complicated. That’s okay. Just like in one dimension, what we’re usually interested in

is the resulting final position and we might call that vector fRuur

and what do you care about in kinematics usually?

Well, one of the things you care about is where it is. Another thing you care about is, how did it travel? What was its change in position? And that would be this vector. This vector represents the change from initial to final. It went that-away. And it’s a vector. It’s got a distance associated with it, a magnitude, but also a direction. It matters which way this arrow is pointing. So, we give this one a name and the traditional name, just like in one dimension, is R∆ . This would be a position vector; this would be a displacement vector. All quantities in two dimensions have analogs. You almost always just use the same name. So, for example, in one dimension, we had this formula , average velocity is

xt

∆∆

. It was a useful quantity. It told you about the change in position with time. What’s the 2-D analog going to be?

What notation should we use? First of all, velocity is going to be a vector. It’s getting to be a little bit of a clumsy notation, but the velocity vector should point in the direction of motion. It should point in the direction of R∆ . The 1-D

formula was xt

∆∆

. The 2-D equation is Rt

∆∆

. It’s the obvious natural definition of velocity. And notice, think about this

equation mathematically. When you take a vector and you divide it by a number, time, that’s multiplying a vector by a

scalar. The scalar is 1t∆

. Multiplying a vector by a scalar does not change direction. It just changes the length. So,

in this case, the velocity vector would be in that direction, but the length of the velocity isn’t exactly the same as the

Kinematics Describing Motion in Two and Three Dimensions The Position and Velocity Vectors Page [2 of 2]

length of the R∆ vector. It’s Rt

∆∆

, the magnitude is the proper word to use for the length of a velocity vector. The

velocity depends on how rapidly you make this motion. It depends on ? t and just like in one dimension, everything we do in 2-D really has a close parallel. You can think about what we did before. You might want to know what happens if my ? t gets very short. If I’m moving along, I might be moving along some complicated path. For instance, here is X and Y and maybe a particle is running along – this is a different kind of graph than I’ve shown you before. It’s not yvst . It’s y v s x . So, to really see the motion, you kind of need an animation. You need to see the object. It’s

walking along its path and as it cruises along, it might be going slow and then fast and then slow. You can’t really see whether it’s moving slow or fast from this picture. You need a little more information. This is motion in 2-D. And I might ask what’s its velocity right here, right now, when it passes this point? What’s its instantaneous velocity? And

we just call that vr

, just like we did before, we just stick an arrow on the top, and it’s going to be the limit as ?t goes to

zero. We want to take the limit for very small times of Rt

∆∆

. And, you know from calculus, the limit of ? goes to zero

of something. It’s just called dr vector divided by dt , it’s drdt

. The derivative of the position tells you the velocity.

So, as the object is moving along, at any instant in time, what direction is the velocity vector? Well, it’s traveling that-

away, so it’s velocity vector must be in that direction and the magnitude is given by drdt

magnitude. If it’s going

quickly here and slowly here, then here you’d use a smaller arrow. The arrow with tangents to this curve and the length of the arrow tells you how rapidly it’s moving. What’s the magnitude of the velocity? We would call the length of this arrow the speed. And now in two dimensions, it’s very clear what’s the distinction between speed and velocity. Speed is just a number, a positive number, which is the size of the velocity vector.

We’ve got equations here, and we’re manipulating things left and right. I’m writing down equations. For instance,

v =rt

∆∆

v. I could rewrite that by multiplying both sides by ?t and then remembering R∆ is fR - iR . fR is iR + v ? t.

You can manipulate these equations in exactly the same way as we did before. In one dimension, we had the

equation fx = ix + v ? t. This is just the two-dimensional extension. Really, all you do is stick arrows over the things

which are vectors. Don’t put an arrow over ?t, that’s not a vector, it’s time, it’s a scalar.

Very abstract. How do I think about these equations? Well, when something is moving in two dimensions, here is the big idea of two-dimensional motion. I can think about the components of the motion. If something is traveling along a path, think about its projection or its shadow in the X direction. As the object moves along, it’s X component moves. As it moves along, it’s also got a Y component. And these equations are vector equations. What does it mean for a vector A to equal to another vector B ? What is that telling me? If two vectors are equal, no matter where they are on the page, they’re still equal, what it means for two vectors to be equal, is xA must be the same as

xB . The X components have to be the same. And, yA has to equal to yB . The Y components have to be the

same. That’s what it means for vectors to be equal. So, how do interpret this equation? I think about components. What’s the X component of this equation? The X component of R is X . fx = ix + the X component of the

average velocity vector times ?t. The X of a vector equation is just a good old one-dimensional kinematics equation that’s familiar. And the Y component is just fY = iY + the Y component of velocity times ? t. As the object moves

around, its velocity vector has an X component and a Y component. This is the X horizontal part of the motion

and this symbol, v Y , represents the vertical component. The principal idea when you’re working in two dimensions, although the symbols look a little bit scarier because they’ve got vector notation. The equations really boil down to one-dimensional motion for the X and one-dimensional motion in the Y world, and you just write two equations and you solve each one separately and independently, and you’ve described the complete two-dimensional motion.