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About this Lesson
- Type: Video Tutorial
- Length: 9:27
- Media: Video/mp4
- Use: Watch Online & Download
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- Download: MP4 (iPod compatible)
- Size: 101 MB
- Posted: 07/02/2009
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This lesson was selected from a broader, comprehensive course, Physics I. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/physics. The full course covers kinematics, dynamics, energy, momentum, the physics of extended objects, gravity, fluids, relativity, oscillatory motion, waves, and more. The course features two renowned professors: Steven Pollock, an associate professor of Physics at he University of Colorado at Boulder and Ephraim Fischbach, a professor of physics at Purdue University.
Steven Pollock earned a Bachelor of Science in physics from the Massachusetts Institute of Technology and a Ph.D. from Stanford University. Prof. Pollock wears two research hats: he studies theoretical nuclear physics, and does physics education research. Currently, his research activities focus on questions of replication and sustainability of reformed teaching techniques in (very) large introductory courses. He received an Alfred P. Sloan Research Fellowship in 1994 and a Boulder Faculty Assembly (CU campus-wide) Teaching Excellence Award in 1998. He is the author of two Teaching Company video courses: “Particle Physics for Non-Physicists: a Tour of the Microcosmos” and “The Great Ideas of Classical Physics”. Prof. Pollock regularly gives public presentations in which he brings physics alive at conferences, seminars, colloquia, and for community audiences.
Ephraim Fischbach earned a B.A. in physics from Columbia University and a Ph.D. from the University of Pennsylvania. In Thinkwell Physics I, he delivers the "Physics in Action" video lectures and demonstrates numerous laboratory techniques and real-world applications. As part of his mission to encourage an interest in physics wherever he goes, Prof. Fischbach coordinates Physics on the Road, an Outreach/Funfest program. He is the author or coauthor of more than 180 publications including a recent book, “The Search for Non-Newtonian Gravity”, and was made a Fellow of the American Physical Society in 2001. He also serves as a referee for a number of journals including “Physical Review” and “Physical Review Letters”.
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Motion in two dimensions is richer than motion in one dimension. You've got more freedom. You can go various directions. That's a good thing. We're trying to describe more complicated physical situations. On the other hand, the price that you pay is that the equations and constants that you write down become a little bit denser. We've defined acceleration to be. It's the same idea that we had all along for acceleration, change in velocity with time. That little arrow on the top is hiding some details. Namely that acceleration is now an arrow. It's got x component and y component. In a certain sense this equation is really two equations. A[x] is dv[x ]dt. And a[y] is dv[y] dt, v[y]. So the same with velocity is derivative of position vector r with respect to time. Again, conceptually I know what I mean by velocity. It's how fast I'm going and which way. It's the change in position with time. And that's what the equation says, and when you think about it like that, it makes sense. Nevertheless, when I look at equations with vectors in them, I must confess to a little bit of unease, because there's all that stuff hidden in them, and I'm afraid that I can't picture it so well. So it's nice to think about ways to picture what these equations are telling us.
One way is very literally to draw pictures. If you have an initial vector v1, v initial, you're going this way fast. It's a big arrow. And then, shortly later, you're going this way, and v final is not so fast. It's a shorter arrow and a new direction. What's the direction of the change? You were going that way. You add the change, and that tells you where you finally ended up. So you can draw pictures, and imagine just shrinking this diagram down to think about the acceleration in this case is in the down and left direction. So that's one nice way of conceptualizing these equations. But ultimately, you want to have as many different ways as you can to think about vector equations in order to really have a good intuition for what they're telling you.
Let's work a problem. A classic kinematics problem would be I describe motion to you. I am going to describe x and y as a function of time, the position vector as a function of time. And then I would like to know, given position as a function of time, what's the velocity as a function of time? What's the acceleration? I'm just trying to describe the aspects of the motion.
So suppose I give you r of t as a formula. How do I do that? Well one way would be to tell you that, at any given moment in time, this vector, which characterizes the position, is some function x of t in the i direction plus some function y of t in the j direction. Here's, remember, the I hat unit vector, and the j hat unit vector. This points in the x direction. That points in the y direction. So if I tell you the functions x of t and y of t, this will localize a point. Let me get very specific. Let me pick an x of t which was five meters per second. That's the number five, and multiply it by time. So I'm going to write some function of time explicitly. I'm just making one up. And let me add five meters per second squared times t^2 in the j direction. This is just being handed to you. This is the description of the motion of some particle. Later we'll see a physical example where this is, in fact, the equation of motion. But for now just consider it to be a formula that's given. This is meters per second multiplied by time gives me meters. Meters per second squared times time squared, again, gives me meters. So the units are all working out. This is the way you describe a position vector as a function of time.
What would this look like in the real world? Let's not worry about it. I'll come back to that. The point is my formulas tell me what's the velocity and acceleration as a function of time mathematically. It's nice just to be able to take these equations and work with them. And you don't have to be able to picture it. It's nice if you can. We'll come back to that.
So let's look at v, velocity as a function of time. It's supposed to be dr by dt. How do you take the derivative of something? Well remember the derivative of the sum is the sum of the derivative, just piece by piece. What's the derivative of this I hat piece? The derivative of five t is just five. So it's just five meters per second in the i hat direction. And then I have to add the derivative of the second term, five meters per second squared t^2. The derivative of t^2 is two t. And multiply two by five. I get ten meters per second squared times time. Units are still okay, because time divided by seconds squared is going to give me meters per second, and I'm after a velocity. So this is the formula for the velocity as a function of time. It's got an x component and a y component. And that means that I know the vector. I know v as a function of time.
How about the acceleration, I just have to take dv dt. Staring at it, what's the derivative of a constant? Nothing, zero. What's the derivative of ten t? It's just ten meters per second squared in the j direction. So, in this case, the acceleration vector is just going to be ten meters per second squared in the j direction.
So I've been given an equation for r and it's mechanical. That's the procedure that you would use to find v and a. Now let's just go back and look at these equations in the physical world. I've drawn up this motion. And just for the fun of it, I have switched my convention on the positive direction for y. It's good practice. Positive y doesn't have to be up the page. It can also be down the page. So this is plus y. And this is plus x. And you remember my formula for position as a function of time. It had a five t with proper units in the x direction. And it had a five t^2 in the j direction. At t equals to zero, I get zero i hat plus zero j hat. J hat is now down the page, and here at t equals to zero. As time marches on, the x component gets bigger, and the y component gets even bigger still, because it's t^2. So that's a parabola. Plug in some times and just march yourself down this graph, and convince yourself that it works out okay. I haven't written down units here just to save myself. When you're doing that, you'll have to see exactly what this tick mark represents, and what these tick marks represent. But the shape of the graph will come up to look like a parabola.
Remember what the x component of the velocity looked like. It was five constant. The x component of the velocity vector is shown in green. What's the real velocity vector as a function of time. Well it's always tangent to the curve. So at this point in time, it would be a vector like so. This is its x piece. That's its y piece. The formula says v[x] is five. V[y] remember was ten T. It starts off small and gets bigger and bigger. And that makes sense. The y component of this tangent is getting steeper and steeper. And the length of this arrow is actually getting bigger and bigger. So I should use a little stubby arrow here that's getting longer and longer and tipping more and more down. That's how I picture this motion. It's consistent, x and y of time and velocity vector as a function of time.
Last equation we got was acceleration, purely y direction. It was ten in the y direction. It was positive. That means that the velocity vector should be changing in the y direction. And you can see that. It's getting bigger and bigger. This represents acceleration in this direction.
So motion in two dimensions, lots of physical situations involve motion in two dimensions, in fact, most applications that we're going to use. And you can write down some lovely compact equations that are the same old equations that we've written down. Definitions, acceleration is change in velocity with time. And these equations describe the motions completely. If you simplify a little bit, for instance, constant acceleration, then we're going to be all set. We're going to be able to write down the equations of motion in two dimensions in terms of the x equations and the y equations. And we'll be able to describe in detail motion for constant acceleration.
Describing Motion in Two and Three Dimensions
Relating Position, Velocity, and Acceleration Vectors in Two Dimensions Page [2 of 2]
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