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About this Lesson
- Type: Video Tutorial
- Length: 11:53
- Media: Video/mp4
- Use: Watch Online & Download
- Access Period: Unrestricted
- Download: MP4 (iPod compatible)
- Size: 127 MB
- Posted: 07/01/2009
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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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You're in the back of a stretch limo, cruising down the highway, nice straight, smooth highway, at high speed, 100 mph. You're sitting in the back, reading a book. It's kind of boring, in fact. Nothing is going on, even though you're going really fast, you don't notice it. You're not paying any attention. What would you notice? You wouldn't notice that you're going fast. But you would certainly notice if your velocity changed. If you're going 100 mph and then a minute later, you're going zero, you wake up, you stop reading the book. Time for a bathroom break. I think the car is coming to a halt. You notice it. And, even more strongly, if you're going down the highway at 100 mph and half a second later, you're going at zero, that means, you were just in a crash and you're probably crunched up against the front windshield.
Physically, what's important in an awful lot of situations is not your velocity, but the change in your velocity. And more importantly, it's not just a change in the velocity, but the change in velocity per unit times, . Going from a 100 to zero in a minute is one thing. Going from 100 to zero in half a second is quite another story. It's such an important physical quantity. It's so relevant in so many physics problems that we've given the ratio a name. And what shall we call it? The very natural English language name. If you're speeding up, we call it acceleration. When you're braking in the car, we call that acceleration. Some people call it deceleration. But we won't use that word. In Physics, it's acceleration and , that's the average acceleration. Remember, I'm talking about the change in velocity over some interval of time. So, I'm finding the average over that chunk of time. And don't forget what D means. D of velocity means V[f] - V[i]. It does not matter what your final velocity or initial velocity is independently, only the change is relevant when you're talking about acceleration. This is the average acceleration and you could look at a simple graph to sort of think a little bit more about what acceleration is and what it tells you. We have been plotting x versus t, but let me plot velocity versus t, now. Because after all, acceleration is given by . If you've got a graph of v versus t, this formula is going to tell you the acceleration. Let's draw that sort of simple, fairly simple graph of v versus t. A straight line. Now, think for a second about what's the motion that this graph is trying to describe. You start at rest and then one second later, you're now moving with a positive velocity. So you start at rest and now you're moving forward and one second later, you're moving forward even faster. This is a graph of something which goes faster and faster and faster. It's accelerating. The formula says that acceleration is . When you look at v versus t graph, you know what that means. Dv divided by Dt is just rise over run on this graph. It's the slope. So if you're given v versus t, you can figure out the average acceleration.
This is a constant acceleration graph because when you have a straight line, a slope is the same wherever you are or whatever time interval, big or small, you use. This is a very common physical situation. Something speeding up or perhaps slowing down, but in a straight line fashion. And we'll work lots of problems where the acceleration is constant. Sometimes, you have to worry about situations where acceleration is not constant. And that's pretty hard to think about. Let me graph v versus t and let me draw one that is not a straight line. So, once again I've decided to start us at rest. After one second, we're going at some fairly small forward velocity. One second later, we're going faster, another second later, we're going faster still. You might think that's the same situation as before, but it's not, because each second, we're going much faster than we were the previous second. So this is an acceleration where somehow it's not staying the same. Your acceleration at the early times is small. Your acceleration at the later times is bigger. How do we work that out? It's very much analogous to what we did when we had x versus t that wasn't a straight line. The idea would be, take , pick a point in time when you're interested in. There's not a common simple acceleration in this picture, so it doesn't make sense to ask, what's the acceleration? You have to say, what's the acceleration now? What's the acceleration at this point in time,.
So, how would you find that? It's the same story as before. You can draw a secant and that slope of that secant would tell you the average acceleration over this long chunk of time and then you could let that long chunk of time get smaller and smaller until finally, you got to the tangent. This is the instantaneous acceleration. It's the slope of the tangent line on the v versus t graph, mathematically. It's limit as Dt goes to zero, of , and of course what that means is instantaneous acceleration is , the derivative of velocity with respect to time. So, mathematically, if I tell you x versus t, you take one derivative to get velocity. If I tell you velocity versus time, mathematically, you take another derivative, in this case, to get to acceleration. So, you can do it graphically; you can do it mathematically. And, figure out instantaneous acceleration.
It's very important to realize that velocity and acceleration are really different things. Just because you know the change in a quantity doesn't mean you know the quantity itself. It's totally disconnected. If you're cruising down the highway in a convertible, 100 mph, it's really exciting. You turn to your friend and you say, we're accelerating like crazy. Your friend knows a little physics and says, "No we're not, we're going 100 mph. It's a steady speed. We're going in a straight line, we're not accelerating at all." It's very easy intuitively to mix up velocity and acceleration. You have to think about the definition every time. If your velocity is not changing, you are not accelerating, no matter how fast you happen to be going at that time.
The units of acceleration are also a little bit funky. I will denote units from time by a square bracket. What are the units of acceleration? It's the units of Dv, that's change in velocity, divided by change in time. Let me give you a specific example. Supposing I'm in a car and I go from zero to 60 mph in twelve seconds. This is my Honda Civic, which doesn't accelerate all that fast, so it's probably a reasonable number. What's the acceleration? Dv is 1-0; it's the difference divided by Dt, that's what this interval represents. 60 12, in this case it would be 5 miles per hour per second. That's the units of acceleration. Kind of weird, in particular it looks especially weird when you convert to metric. Miles per hour has to be converted to meters per second and so it becomes meters per second per second, or meters per second squared. It's a combined unit. That's what it is. You plug it into formulas. It's okay.
When you look at this number, it makes sense to me. Five miles per hour each second. After twelve seconds, I have changed my velocity by 12 x 5 = 60 miles per hour. So, units are a little funky, but they're correct. The toughest part about acceleration, the toughest part to, I think intuit, is the sign. If you got the formulas, you can always work it out, but if you're just thinking about some motion, sometimes it's a little bit awkward, thinking about the sign of the acceleration. Let me give you an example. Here is little iguanadon sitting at rest at the origin and it starts to accelerate towards the right. Right is positive x. That's my convention for signs on the coordinate system. So it starts off slowly and then faster, and faster, and faster and pretty soon is racing off to the right. So, off the screen. Going faster and faster and faster to the right, what's the sign of the acceleration? Well, that one's pretty easy. Acceleration is and Dv is V[f] - V[i]. V[f] is a big number to the right. was zero, so you've got a big positive number minus zero. That's a positive change. Positive acceleration means you're going faster and faster in the right hand direction. That's okay. Here's a slightly funkier one. Suppose that you are coming from the right, fast and you put on the brakes and slow down and come to a halt. So, your initial velocity was to the left. V[i] is a big negative number. V[f] is zero. So you take zero and you subtract a big negative number. Subtracting a negative is positive. Dv is positive again. This is a situation of positive acceleration. Even though you were moving to the left, you were slowing down and that's an acceleration in this direction. Let me draw you a little picture. Sometimes this helps people. Here was my initial velocity - negative, big. Here was my final velocity, basically zero. What's the change? What do you have to add to your initial velocity to end up at the final? Here is Dv. You have to add an arrow to the right. Let me do one more example. It's useful to just kind of think through a few examples, and after awhile, you can get used to this idea of the sign of the acceleration. Supposing that I come in from the left side, moving fast and come to a halt. If I was moving rapidly to the right, slower, slower, slower, and I come to rest. So my initial velocity was big and positive, my final velocity was zero, Dv is zero minus a positive number, negative. If you're coming in this way and slowing down, the change is that way. It's a negative acceleration. Most people find that reasonably intuitive. If you're moving fast and hitting the brakes, calling that a negative acceleration seems quite reasonable.
So, acceleration is . You can talk about instantaneous or average accelerations and mathematically it is straightforward. It's the slope of velocity versus time , or if you prefer, the derivative, . So, it's straightforward to find. People's intuitions about it, it's a little too easy to mix up acceleration and velocity. So you just kind of have to stare at the formulas a little bit.
In Physics, what we're going to discover over and over again, is velocity isn't really such an important physical quantity. Accelerations are what's important. When we talk about forces, when we talk about Newton's laws, we're going to see all the time, it's the change in velocity. It's the acceleration that's really interesting. That's where all the physics is. That's where the action is, is in the acceleration.
Investigating One-Dimensional Motion
Acceleration Page [1 of 3]
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