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About this Lesson
- Type: Video Tutorial
- Length: 9:02
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- 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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There's lots of times in Physics when average velocity is really all you care about. For instance, when iguanadon runs a little race, the important thing is, what's the average velocity? What's the ? Remember, if you're running a race, Dx is fixed and you just want to make Dt as small as possible to make your average velocity as big as possible. That's all you care about.
On the other hand, if this was a road race, and it was on city streets, it might be case that iguanadon speeds up and is going really fast, and then maybe gets tired and runs out of gas and slows down. If there's a cop standing at the side of the road and sees that, at this point, iguanadon was going at 80 miles an hour, iguanadon is busted. And it's not going to help if iguanadon says, "Yeah, by my average velocity was only 55." In this case, you care about something different than the average over the whole trip. You care about how fast were you going now, at some moment in time.
What could we mean precisely, by the velocity at an instant in time. We've defined velocity to be the average over some interval of time. Let me give you another physical example to talk about this. When you drop a ball, it starts at rest, and it goes faster and faster as it falls. And we've seen that you can talk about the average velocity for the whole trip and you can also ask, what was the average velocity in this part of the trip? What was the average velocity in that part? They'll all be different. Supposing I want to ask the following question. At t=1. So I start my stopwatch and the ball is falling, and right here at this instant, how fast is it going? It's a concept that when you're in your car makes total sense. If you want to know how fast are you going, you look down at the speedometer. There's a needle and it tells you, how fast am I going right now? The needle changes with time. Now, I'm going faster, now I'm going slower. And in the end, your speedometer doesn't tell you your average speed. You have to look at your odometer, the thing that tells you how far you went and you look at your watch and you say for the average of the trip.
Well, we don't have a speedometer built into this thing. So, how could we figure out, using kinematics, what is its instantaneous velocity? That's the word I like to use for the velocity at an instant in time. It's a slightly funny concept and let's go back to the graph of position vs. time, which is kind of our basic way of describing any kind of motion, at least to start with. So, when you drop the ball, it starts at x=0. We're defining down to be positive in this case. So, as time marches along, at one second, you're at this x, at two seconds, you're at a bigger x, and at any moment in time, this graph, you just read it off, where are you? We talked about how you can figure out your average velocity. For instance, from t=1 seconds, to this point, 4 seconds - I haven't labeled my axis, but it's implicit here. The way you find the average velocity is, you lay down a secant. Average velocity is ; Dt is the run of this line, and Dx would be the rise of this line, and you have to start from this point right here. Now, if I want to know what's my instantaneous velocity at one second. If that's the question that I'm interested in - I'm the cop - and at one second, I want to check your speed, your velocity, I really don't want to average over the next three seconds. Why three seconds? Why not average over the next two seconds? At two seconds, notice that Dx is different, Dt is different, and when you look at the picture, you can see that the slope of the secant is smaller. The average over these two seconds, the average velocity is a smaller number. And, over one second, it's even smaller still. It's a smaller number and you have to look at the slope of the secant to see that. It's the slope of the secant that tells you the average velocity. I don't want to average over four seconds, or three or two. I don't even want to average over a tenth of a second. I want to know the average over an instant. Well, that's a pretty strange concept and in calculus, you spend a lot of time really defining exactly what you mean by in the limit that you take Dt to be a really, really tiny number. In fact, you really take the limit all the way down to t=0. That's the mathematical definition of instantaneous velocity. Limit Dt goes to zero at . It's fancy mathematics, calculus. But when you look at the graph, you can see what we want. We want to take the slope of the secant line in the limit that we just take a Dt that's really, really small. When you look at the graph, what you want is the slope of the tangent line, as the secant progressively slides down and down like this, I'm basically sliding the contact point, the final time that I'm average over when it gets to very, very close to t initial, what you end up with is a tangent line.
Another way of thinking about this is, if you're zooming in on the graphs, and you're looking more and more closely at the instant t=1, even a curvy graph, as you zoom in on it, begins to look more and more straight, and in the limit that you've blown it up sort of infinitely, the slope of that straight line is exactly the slope of the tangent line.
So, that's the formula for velocity and if you look at the graph, you can usually just stare at it and see, immediately, the velocity is small here. The slope of this line is small. The velocity at t=2, that's the tangent line here, is bigger and bigger and bigger. The tangent line is getting steeper and steeper and it's going faster and faster, which makes sense. As you drop the ball, it's going faster and faster. You can have an arbitrarily complicated graph, x vs. t, might look like, I don't know, something like that. And if you've got the graph in front of you, you can figure out what's the velocity at, for instance, at this time, t1, you go up and you ask, what's the slope of the tangent right there? That's the velocity. What's the velocity at ? It's zero. Does that make sense? The slope of this line is zero, rise over run is nothing. Well, think about the motion. It's moving forward, x is increasing with time. At this moment in time, it's stopped moving forward. It's about to move backwards. This is a graph of a particle which is moving forwards, comes to a halt and moves backward, and at the moment it comes to a halt, that's , its velocity, its instantaneous velocity is zero. What's happening next? It has come to a stop, it is now moving backwards. Dx is now negative. It's moving to the left, it's moving in the opposite direction. And, at some time, say here, at t3, you can see that the slope is negative. When a line is tipped down and right, rise over run is really fall over run, it's a negative slope, a negative velocity. Remember, negative velocity just tells you direction. It means you're heading the other way, away from the direction of positive x.
And that's the graphical way of measuring and thinking about instantaneous velocity. By the way, lot's of people will leave off the adjective instantaneous, and just say, velocity. What's the velocity of a particle? In general, if you don't add an adjective average, it's assuming that you're always talking about instantaneous velocity. In an awful lot of physics problems, you really want to know velocity, instantaneous velocity is a function of time. It's a really useful quantity and it's easy to get, at least approximately, from the graph, and what we'd we really like to do, what we'll talk about next, is how to get it quantitatively. If you really want to take the functional form of this graph and derive a formula for instantaneous velocity.
Investigating One-Dimensional Motion
Understanding Instantaneous Velocity Page [1 of 2]
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