Physics: Fluid Motion: Drag Force, Terminal Speed

This lesson has not been reviewed.

Please purchase the lesson to review.

Sliding friction between two surfaces isn't the only kind of friction in the world. If you've ever ridden a bicycle, you've surely noticed that, as you go faster and faster, there is more and more air resistance. That's a kind of frictional force. It's resisting your motion. If you turn around and go backward, the force of friction is never helping you. Air resistance is always holding you back. It's a force of nature, and if we want to work problems that involve objects moving through air, we would certainly like to have a formula for the drag force.
So drag force, it's a kind of a friction. And what could it depend on? I'm going to write down a formula for you, but just like the formula for sliding friction, you should bear in mind that it's an approximate formula. I'm going to make some simplifying approximations, and it's not always true. I'll talk a little bit about circumstances where the formula breaks down. It's a crude formula that really comes from experiment. People make measurements of drag forces and come up with what does it depend on?
So you ask yourself, when we had sliding friction, the friction force depended really only on the normal force, the contact force. When you're moving through air, I use the word fluid like a physicist, meaning air or liquid, gas, just some object moving through a fluid will feel a drag. And there is no obvious normal force. I mean there are some molecules. Microscopically the molecules in the fluid are bouncing against me, and that's got to be the origin of this drag force. But when I think about what the drag force could depend on, I ask, for instance, does it depend on speed? Sliding friction does not depend on speed. That's experimentally observable. Drag force does. It depends on speed. The faster you go, the more resistance you feel. And if you go into the laboratory, you discover that it's worse than proportional to v. It goes like the square of the velocity. So if you double your speed, you get four times the air resistance or air drag. So here's already an approximation. This power of two isn't true in all circumstances. If you're going really slowly, and then you double your speed, the drag will not go up by a factor of four. At low speeds, drag force, in fact goes like v, just v, not v squared. Even that depends on what fluid you're going through. When you're going through air at moderate speeds, toss a tennis ball or something, this formula is okay. It's pretty good.
What else does drag depend on? It depends on the density. This is the symbol rho. It's the mass per unit volume of the stuff, the fluid that you're going through. So if you have dense gas that you're trying to go through, you will have more drag. That makes sense from this microscopic picture, I think.
Does it depend on anything else? Well, you've kind of got to scratch your head and think about it a little bit. If you're on your bicycle and you open up your jacket, clearly there's going to be a lot more drag force. So what is that depending on? I think it's the cross-sectional area, the area that you're presenting in the direction of travel. So we write it's proportional to the area.
These are pretty much the important physical quantities that drag force depends on. It's proportional to all three, so let's just write down a formula for drag force now. It's half C, that half is purely conventional. This is my proportionality constant. It depends on the density of the fluid that you're going through, the cross-sectional area you present, and the square of the velocity. This formula tells you what you need to know for approximate problem solving. Once you know a formula for the force, you can solve Newton's second law problems and draw the arrow. It's going to oppose the motion. Here's the length.
Let's look at the units of this equation. Force has units of kilograms meters per second squared. That's the units of Newtons. So let me not worry about the C yet. Let's just look at the physical quantities here, the ones that are sort of what it depends on. Density is mass per volume. That's the symbol rho. It looks like a P here, but it's really the Greek letter rho. It's kilogram per cubic meter, mass per volume. A is cross-sectional area, that's meters squared. And velocity is meters per second. And we're squaring that. So I've got one, two, three, four Ns upstairs and three downstairs. So that's one left upstairs. I've got kilograms meters per second squared. The units match. That's nice. It tells us that this coefficient is unitless, it's just a number.
What could it depend on? It depends, I suppose, primarily on the texture. What's the surface made of? Some sort of details of the object involved. It's kind of an interesting fact that fuzzy things tend to have a somewhat smaller, this is called the coefficient of drag, and coefficient of drag tends to be a little bit smaller for fuzzy things. That's why tennis balls are fuzzy, and golf balls are dimpled. The physics there is very subtle. It has to do with the fact that for this formula to apply, you really need some turbulence behind the object. If the object is like an arrow, or very streamlined, this formula doesn't work very well. And if the velocities are very small, so you that you don't get turbulence again, this formula doesn't work so well. The power becomes v. You've got to come up with some other formula.
But this formula will work in lots of reasonably common, physical situation. Let's do an example. Supposing that you have jumped out of an airplane. So here's a parachutist, jumps out of an airplane. And you kind of know what the behavior is going to look like. You watch this thing falling. I'd like to analyze the motion. I'd like to really say in detail, is it accelerating downward, but less than g? Or is it moving down with a constant velocity? What's it doing?
It actually depends. It's a function of time. When you first jump out of the airplane, first of all you haven't opened your chute yet, so here's the free-body diagram. Here's you. And you've jumped out of the airplane. Let's assume that you just stepped out from a helicopter. So zero velocity at first. No velocity means that the drag formula says no velocity, no drag force, not at first. So you've only got your weight, mg, down. So let me call down positive. And Newton's law says the sum of the forces--that's it. It's the only force. That's mg. Sum of the forces should equal to ma. And cancel the mass, your acceleration is equal to g. Of course, if you step out of the airplane, there's no drag force yet. Galileo tells us that everything, no matter what, will accelerate 9.8 meters per second squared down. Constant acceleration, so we can use our kinematic formulas for constant acceleration, which is zero, plus a, which is g, times T.
So your velocity is steadily increasing, faster and faster, straight line. If I were to graph downward velocity versus time, it starts off increasing like a straight line. Will it keep increasing forever, and you'll go faster and faster and faster? Surely not, because the faster you go, the bigger the drag force gets. As you increase your speed, when you draw the free-body diagram, mg is always there down. And the drag force is getting bigger and bigger. It's got that formula. It will continue to get bigger as you go faster and faster until there comes a moment when these two arrows are equal. What happens then? At that moment, at that speed, drag force is equal to the downward gravitational force. So if I write the sum of the forces equals ma, I've got mg down minus f drag up. And if these things are balanced, the answer is zero, no more acceleration. Does that mean you come to a halt? Of course not, you're traveling downward at some velocity, some reasonably high velocity, and it's just not changing anymore. There is now no net force on you. So that's what you're seeing here. This thing has reached its terminal velocity. That's the name for that speed when the drag force has gotten to equal weight. And from then on it moves down with constant speed.
We can use this formula to solve for that terminal velocity. Terminal means the final velocity. Just set fd equal to mg. So I have , that's the drag force, is equal to the weight. And you can solve that easily for velocity. It's . So that's a nice formula. It's the maximum speed that an object will get to while it's falling in the presence of both gravity and air drag. And you can see, for example, why you want a parachute. If you put a big area up, then it's in the denominator, and so that will decrease your terminal velocity. It seems reasonable. And you can plug in numbers and figure out what the terminal velocity of a person is. You just need to know what the C coefficient is. It's something that you either have to measure or look up in a reference somewhere.
Let me draw you a picture of... There it is. Here's my graph for short times. What happens for large times? At large times, I said that we've reached some terminal velocity. So the graph has to go from this to that. And it's going to do it in a smooth way. The details of this curve are a little bit tricky, but as a function of time, this is a graph of your downward velocity. It starts off increasing linearly. And then, at a certain point, this is asymptotic. It's approaching that horizontal line. And at large times, you're basically going down with essentially a constant terminal velocity.
Friction in problems can always be added. Unfortunately, our formulas are a little bit approximate. But as long as the conditions are reasonable, you can just use our formulas for sliding friction, or for drag friction. And just use Newton's second law, and figure out what's going on. You need a few details in any friction problem. I wish I could give you a rule of thumb so you would just know when do I need to worry about friction, and when can I neglect it? There is no such rule of thumb. You have to just look at the problem, and think about it a little bit. Certainly much of the time, I'm a physicist, I want to keep my life as simple as possible. Many problems, I'll think a little bit about friction, and say, "Ah, forget about it. It's small. Let's just completely neglect it." But if you're dropping out of an airplane, you certainly can't neglect air drag. And you can just compute with Newton's second law like we always do.
Dynamics
The Forces of Friction
Motion Through a Fluid: Drag Froce and Terminal Speed Page [3 of 3]