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What are the different applications of terminal velocity?

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Answer: Hm... *applications*, as opposed to just "terminal velocity happens"? Parachutes. Hang gliders. If you're a mouse or smaller, you're at one end of the squares-cubes law where you have a lot of surface area for your mass, and thus terminal velocity is low enough that you can't get seriously...

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Terminal velocity, steady speed achieved by an object freely falling through a gas or liquid. A typical terminal velocity for a parachutist who delays opening the chute is about 150 miles (240 kilometres) per hour. Raindrops fall at a much lower terminal velocity, and a mist of tiny oil droplets settles at an exceedingly small terminal velocity. An object dropped from rest will increase its speed until it reaches terminal velocity; an object forced to move faster than its terminal velocity will, upon release, slow down to this constant velocity.

Terminal velocity is achieved, therefore, when the speed of a moving object is no longer increasing or decreasing; the object's acceleration (or deceleration) is zero. The force of air resistance is approximately proportional to the speed of the falling object, so that air resistance increases for an object that is accelerating, having been dropped from rest until terminal velocity is reached. At terminal velocity, air resistance equals in magnitude the weight of the falling object. Because the two are oppositely directed forces, the total force on the object is zero, and the speed of the object has become constant.

The Editors of Encyclopaedia Britannica Adam Augustyn, Managing Editor, Reference Content.

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terminal velocity | Definition, Examples, & Facts

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Terminal velocity, steady speed achieved by an object freely falling through a gas or liquid. An object dropped from rest will increase its speed until it reaches terminal velocity; an object forced to move faster than it terminal velocity will, upon release, slow down to this constant velocity.

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Terminal velocity is the maximum velocity attainable by an object as it falls through a fluid (air is the most common example). It occurs when the sum of the drag force () and the buoyancy is equal to the downward force of gravity () acting on the object. Since the net force on the object is zero, the object has zero acceleration.

In fluid dynamics, an object is moving at its terminal velocity if its speed is constant due to the restraining force exerted by the fluid through which it is moving .

As the speed of an object increases, so does the drag force acting on it, which also depends on the substance it is passing through (for example air or water). At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). At this point the object ceases to accelerate and continues falling at a constant speed called the terminal velocity (also called settling velocity). An object moving downward faster than the terminal velocity (for example because it was thrown downwards, it fell from a thinner part of the atmosphere, or it changed shape) will slow down until it reaches the terminal velocity. Drag depends on the projected area, here, the object's cross-section or silhouette in a horizontal plane. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a bullet. In general, for the same shape and material, the terminal velocity of an object increases with size. This is because the downward force (weight) is proportional to the cube of the linear dimension, but the air resistance is approximately proportional to the cross-section area which increases only as the square of the linear dimension. For very small objects such as dust and mist, the terminal velocity is easily overcome by convection currents which prevent them from reaching the ground and hence they stay suspended in the air for indefinite periods. Air pollution and fog are examples of this.

Based on wind resistance, for example, the terminal speed of a skydiver in a belly-to-earth (i.e., face down) free fall position is about 195 km/h (120 mph; 54 m/s). This speed is the asymptotic limiting value of the speed, and the forces acting on the body balance each other more and more closely as the terminal speed is approached. In this example, a speed of 50% of terminal speed is reached after only about 3 seconds, while it takes 8 seconds to reach 90%, 15 seconds to reach 99% and so on.

Higher speeds can be attained if the skydiver pulls in his or her limbs (see also freeflying). In this case, the terminal speed increases to about 320 km/h (200 mph or 90 m/s), which is almost the terminal speed of the peregrine falcon diving down on its prey. The same terminal speed is reached for a typical .30-06 bullet dropping downwards-when it is returning to the ground having been fired upwards, or dropped from a tower-according to a 1920 U.S. Army Ordnance study.

Competition speed skydivers fly in a head-down position and can reach speeds of 530 km/h (330 mph; 150 m/s); the current record is held by Felix Baumgartner who jumped from a height of 128,100 feet (39,000 m) and reached 1,357.6 km/h (840 mph; 380 m/s), though he achieved this speed at high altitude, where extremely thin air presents less drag force.

The biologist J. B. S. Haldane wrote,

To the mouse and any smaller animal [gravity] presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object.

Using mathematical terms, terminal speed-without considering buoyancy effects-is given by

where

represents terminal velocity,

In reality, an object approaches its terminal speed asymptotically.

Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass has to be reduced by the displaced fluid mass , with the volume of the object. So instead of use the reduced mass in this and subsequent formulas.

The terminal speed of an object changes due to the properties of the fluid, the mass of the object and its projected cross-sectional surface area.

Air density increases with decreasing altitude, at about 1% per 80 metres (260 ft) (see barometric formula). For objects falling through the atmosphere, for every 160 metres (520 ft) of fall, the terminal speed decreases 1%. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed.

Derivation for terminal velocity

Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation):

with v( t) the velocity of the object as a function of time t.

Derivation of the solution for the velocity v as a function of time t

The drag equation is-assuming ρ, g and C d to be constants:

Although this is a Riccati equation that can be solved by reduction to a second-order linear differential equation, it is easier to separate variables.

Dividing both sides by m gives

The equation can be re-arranged into

Taking the integral of both sides yields

After integration, this becomes

or in a simpler form

with arctanh the inverse hyperbolic tangent function.

Alternatively,

with tanh the hyperbolic tangent function. Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes

As time tends to infinity ( t → ∞ ), the hyperbolic tangent tends to 1, resulting in the terminal speed

Terminal speed in a creeping flow

For very slow motion of the fluid, the inertia forces of the fluid are negligible (assumption of massless fluid) in comparison to other forces. Such flows are called creeping flows and the condition to be satisfied for the flows to be creeping flows is the Reynolds number, . The equation of motion for creeping flow (simplified Navier-Stokes equation) is given by

where

is the fluid velocity vector field,

is the fluid pressure field,

The analytical solution for the creeping flow around a sphere was first given by Stokes in 1851. From Stokes' solution, the drag force acting on the sphere can be obtained as

where the Reynolds number, . The expression for the drag force given by equation (6) is called Stokes' law.

When the value of is substituted in the equation (5), we obtain the expression for terminal speed of a spherical object moving under creeping flow conditions:

where is the density of the object.

Applications

The creeping flow results can be applied in order to study the settling of sediments near the ocean bottom and the fall of moisture drops in the atmosphere. The principle is also applied in the falling sphere viscometer, an experimental device used to measure the viscosity of highly viscous fluids, for example oil, paraffin, tar etc.

When the buoyancy effects are taken into account, an object falling through a fluid under its own weight can reach a terminal velocity (settling velocity) if the net force acting on the object becomes zero. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. That is

where

= weight of the object,

= buoyancy force acting on the object, and

= drag force acting on the object.

If the falling object is spherical in shape, the expression for the three forces are given below:

where

is the diameter of the spherical object,

is the gravitational acceleration,

is the density of the fluid,

is the density of the object,

is the projected area of the sphere,

is the drag coefficient, and

is the characteristic velocity (taken as terminal velocity, ).

Substitution of equations (2-4) in equation (1) and solving for terminal velocity, to yield the following expression

In equation (1), it is assumed that the object is denser than the fluid. If not, the sign of the drag force should be made negative since the object will be moving upwards, against gravity. Examples are bubbles formed at the bottom of a champagne glass and helium balloons. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up.

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Terminal velocity is the maximum velocity attainable by an object as it falls through a fluid (air is the most common example). It occurs when the sum of the drag force () and the buoyancy is equal to the downward force of gravity () acting on the object.

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OpenStaxCollege

Learning Objectives

Express mathematically the drag force.

Discuss the applications of drag force.

Define terminal velocity.

Determine the terminal velocity given mass.

Another interesting force in everyday life is the force of drag on an object when it is moving in a fluid (either a gas or a liquid). You feel the drag force when you move your hand through water. You might also feel it if you move your hand during a strong wind. The faster you move your hand, the harder it is to move. You feel a smaller drag force when you tilt your hand so only the side goes through the air-you have decreased the area of your hand that faces the direction of motion. Like friction, the drag force always opposes the motion of an object. Unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid. This functionality is complicated and depends upon the shape of the object, its size, its velocity, and the fluid it is in. For most large objects such as bicyclists, cars, and baseballs not moving too slowly, the magnitude of the drag force is found to be proportional to the square of the speed of the object. We can write this relationship mathematically as . When taking into account other factors, this relationship becomes

where is the drag coefficient, is the area of the object facing the fluid, and is the density of the fluid. (Recall that density is mass per unit volume.) This equation can also be written in a more generalized fashion as , where is a constant equivalent to . We have set the exponent for these equations as 2 because, when an object is moving at high velocity through air, the magnitude of the drag force is proportional to the square of the speed. As we shall see in a few pages on fluid dynamics, for small particles moving at low speeds in a fluid, the exponent is equal to 1.

Athletes as well as car designers seek to reduce the drag force to lower their race times. (See [link]). "Aerodynamic" shaping of an automobile can reduce the drag force and so increase a car's gas mileage.

The value of the drag coefficient, , is determined empirically, usually with the use of a wind tunnel. (See [link]).

The drag coefficient can depend upon velocity, but we will assume that it is a constant here. [link] lists some typical drag coefficients for a variety of objects. Notice that the drag coefficient is a dimensionless quantity. At highway speeds, over 50% of the power of a car is used to overcome air drag. The most fuel-efficient cruising speed is about 70-80 km/h (about 45-50 mi/h). For this reason, during the 1970s oil crisis in the United States, maximum speeds on highways were set at about 90 km/h (55 mi/h).

Typical values of drag coefficient .

Object

C

Airfoil

0.05

Toyota Camry

0.28

Ford Focus

0.32

Honda Civic

0.36

Ferrari Testarossa

0.37

Dodge Ram pickup

0.43

Sphere

0.45

Hummer H2 SUV

0.64

Skydiver (feet first)

0.70

Bicycle

0.90

Skydiver (horizontal)

1.0

Circular flat plate

1.12

Substantial research is under way in the sporting world to minimize drag. The dimples on golf balls are being redesigned as are the clothes that athletes wear. Bicycle racers and some swimmers and runners wear full bodysuits. Australian Cathy Freeman wore a full body suit in the 2000 Sydney Olympics, and won the gold medal for the 400 m race. Many swimmers in the 2008 Beijing Olympics wore (Speedo) body suits; it might have made a difference in breaking many world records (See [link]). Most elite swimmers (and cyclists) shave their body hair. Such innovations can have the effect of slicing away milliseconds in a race, sometimes making the difference between a gold and a silver medal. One consequence is that careful and precise guidelines must be continuously developed to maintain the integrity of the sport.

Some interesting situations connected to Newton's second law occur when considering the effects of drag forces upon a moving object. For instance, consider a skydiver falling through air under the influence of gravity. The two forces acting on him are the force of gravity and the drag force (ignoring the buoyant force). The downward force of gravity remains constant regardless of the velocity at which the person is moving. However, as the person's velocity increases, the magnitude of the drag force increases until the magnitude of the drag force is equal to the gravitational force, thus producing a net force of zero. A zero net force means that there is no acceleration, as given by Newton's second law. At this point, the person's velocity remains constant and we say that the person has reached his terminal velocity ( ). Since is proportional to the speed, a heavier skydiver must go faster for to equal his weight. Let's see how this works out more quantitatively.

At the terminal velocity,

Thus,

Using the equation for drag force, we have

Solving for the velocity, we obtain

Assume the density of air is . A 75-kg skydiver descending head first will have an area approximately and a drag coefficient of approximately . We find that

This means a skydiver with a mass of 75 kg achieves a maximum terminal velocity of about 350 km/h while traveling in a pike (head first) position, minimizing the area and his drag. In a spread-eagle position, that terminal velocity may decrease to about 200 km/h as the area increases. This terminal velocity becomes much smaller after the parachute opens.

Find the terminal velocity of an 85-kg skydiver falling in a spread-eagle position.
Strategy
At terminal velocity, . Thus the drag force on the skydiver must equal the force of gravity (the person's weight). Using the equation of drag force, we find .

Thus the terminal velocity can be written as
Solution
All quantities are known except the person's projected area. This is an adult (82 kg) falling spread eagle. We can estimate the frontal area as

Using our equation for , we find that
Discussion
This result is consistent with the value for mentioned earlier. The 75-kg skydiver going feet first had a . He weighed less but had a smaller frontal area and so a smaller drag due to the air.

The size of the object that is falling through air presents another interesting application of air drag. If you fall from a 5-m high branch of a tree, you will likely get hurt-possibly fracturing a bone. However, a small squirrel does this all the time, without getting hurt. You don't reach a terminal velocity in such a short distance, but the squirrel does.

The following interesting quote on animal size and terminal velocity is from a 1928 essay by a British biologist, J.B.S. Haldane, titled "On Being the Right Size."
To the mouse and any smaller animal, [gravity] presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, and a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object. Divide an animal's length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistance to falling in the case of the small animal is relatively ten times greater than the driving force.
The above quadratic dependence of air drag upon velocity does not hold if the object is very small, is going very slow, or is in a denser medium than air. Then we find that the drag force is proportional just to the velocity. This relationship is given by Stokes' law, which states that

where
is the radius of the object,
is the viscosity of the fluid, and
is the object's velocity.

Good examples of this law are provided by microorganisms, pollen, and dust particles. Because each of these objects is so small, we find that many of these objects travel unaided only at a constant (terminal) velocity. Terminal velocities for bacteria (size about
) can be about
. To move at a greater speed, many bacteria swim using flagella (organelles shaped like little tails) that are powered by little motors embedded in the cell. Sediment in a lake can move at a greater terminal velocity (about
), so it can take days to reach the bottom of the lake after being deposited on the surface.

If we compare animals living on land with those in water, you can see how drag has influenced evolution. Fishes, dolphins, and even massive whales are streamlined in shape to reduce drag forces. Birds are streamlined and migratory species that fly large distances often have particular features such as long necks. Flocks of birds fly in the shape of a spear head as the flock forms a streamlined pattern (see [link]). In humans, one important example of streamlining is the shape of sperm, which need to be efficient in their use of energy.

Galileo is said to have dropped two objects of different masses from the Tower of Pisa. He measured how long it took each to reach the ground. Since stopwatches weren't readily available, how do you think he measured their fall time? If the objects were the same size, but with different masses, what do you think he should have observed? Would this result be different if done on the Moon?

PhET Explorations: Masses & Springs

A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.

The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meters per second and kilometers per hour) of an 80.0-kg skydiver falling in a pike (headfirst) position with a surface area of .

A 560-g squirrel with a surface area of falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a horizontal skydiver.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance?

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Express mathematically the drag force. Discuss the applications of drag force. Define terminal velocity. Determine the terminal velocity given mass. Another interesting force in everyday life is the force of drag on an object when it is moving in a fluid (either a gas or a liquid).

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Discussion

pressure drag

The force on an object that resists its motion through a fluid is called drag. When the fluid is a gas like air, it is called aerodynamic drag or air resistance. When the fluid is a liquid like water it is called hydrodynamic drag, but never "water resistance".

Fluids are characterized by their ability to flow. In somewhat technical language, a fluid is any material that can't resist a shear force for any appreciable length of time. This makes them hard to hold but easy to pour, stir, and spread. Fluids have no definite shape but take on the shape of their container. (We'll ignore surface tension for the time being. It's really only significant on the small scale - small like the size of a drop.) Fluids are polite in a sense. They yield their space relatively easily to other material things; at least when compared to solids. A fluid will get out of your way if you ask it. A solid has to be told to get out of the way with destructive force.

Fluids may not be solid, but they are most certainly material. The essential property of being material (in the classical sense) is to have both mass and volume. Material things resist changes in their velocity (this is what it means to have mass) and no two material things may occupy the same space at the same time (this is what it means to have volume). The portion of the drag force that is due to the inertia of the fluid - the resistance that it has to being pushed aside - is called the pressure drag (or form drag or profile drag). This is usually what someone is referring to when they talk about drag.

Recall Bernoulli's equation for the pressure in a fluid...

The first term on each side of the equation is the part of the pressure that comes from outside the fluid. Typically, this refers to atmospheric pressure weighing down on the surface of a liquid (not relevant right now). The second term is the gravitational contribution to pressure. This is what causes buoyancy (also not relevant right now). The third term is the kinetic or dynamic contribution to pressure - the part related to flow (very relevant right now). This will help us understand the origin of pressure drag.

Start with the definition of pressure as force per area. Solve it for force.

Replace the generic symbol F for force with the more specific symbol R for drag. (You could also use D if you wanted to.) Drop in Bernoulli's equation for the pressure in a moving fluid...

Rearrange things a bit and here you go...

Wait a minute. Where'd that extra symbol come from? Who put that C in there and why?

Let's run through all the symbols one at a time, explain their meaning and how they relate to pressure drag. In essence, let's take the equation apart and put it back together again.

Drag increases with the density of the fluid (ρ). More density means more mass, which means more inertia, which means more resistance to getting out of the way. The two quantities are directly proportional.
R ∝ ρ

Drag increases with area (A). Exactly what we mean by this is subject to debate. To me, and in the context of this model, area is the cross sectional area projected in the direction of motion. (I would further simplify this by calling it the projected area.) Take the cross section of the object in the direction of its motion. This is the area of the tube of fluid that must be cast aside to let the object pass. This is the most logical thing to call the area, but not everyone agrees with me. To some, the word "area" refers to the area of contact between the object and the fluid. This also makes sense, but not in the context I've described above. Surface area is not important when one is dealing with pressure drag, but it is important when dealing with viscous drag - drag caused by layers of the fluid sticking to the object and to one another. More surface area means more of the object is in contact with the fluid, which means more drag. Viscous drag is just as real as pressure drag, but I don't want to deal with it right now.

Drag increases with speed (v). I hope that this is self-evident. An object that is stationary with respect to the fluid will certainly not experience any drag force. Start moving and a resistive force will arise. Get moving faster and surely the resistive force will be greater. The hard part of this relationship lies in the detailed way speed affects drag. According to our sensible model derived from Bernoulli's sensible equation, drag should sensibly be proportional to the square of speed. In some situations, however, this may not be entirely correct. Drag is a complex phenomena. It cannot always be described with equations that are simple. My first guess would always be that drag is proportional to the square of speed since I understand and like the derviation I've presented, but I would not be surprised if (over some range of values) drag and speed were found to be directly proportional, proportional to some power other than 2, or related by some polynomial. Welcome to the world of empirical modeling - where relationships are determined by actual physical experiments rather than an ideology of pure theory. More on this subject later in this section.

Which brings us to our last factor...

Drag is influenced by other factors including shape, texture, viscosity (which results in viscous drag or skin friction), compressibility, lift (which causes induced drag), boundary layer separation, and so on. These factors can be dealt with separately in a more complete theory of drag (how tedious in one sense, but how necessary in another) or they can be piled into one monolithic fudge factor (oh yes, please) called the coefficient of drag (C).

Combining all these factors together yields a theoretically limited (but empirically reasonable) equation. Here it is again...

Simple, compact, wonderful. A nice equation to work with - or is it?

Well, yes and no.

1. Yes, but it works only as long as the range of conditions examined is "small". That is, no large variations in speed, viscosity, or crazy angles of attack. The way around this is to reduce the coefficient of drag to a variable rather than a constant. (I can live with this.) Say that C depends on some yet to be specified set of factors. It is totally acceptable to say that it varies with this that or the other quantity according to any set of rules determined by experiment.

2. No, since speed is squared. [Gasp!] Recall that speed is the derivative of distance with respect to time. Have you ever tried to solve a nonlinear differential equation? No? Well, welcome to hell. Wait, let me rephrase that - Welcome to Hell! [Ca-rack! Boom!] Ah ha ha ha ha haaaa! [Rumble] You fool! Just wait till you see what's in store for you when you try to solve the differential equations. The mathematics will consume you. [Ca-rack! Boom!] Ah ha ha ha ha haaaa! [Rumble].

Whew. What the hell was that all about? I might not know how to solve every kind of differential equation off the top of my head, but so what. I can always look for the solution in a book of standard mathematical tables or an on-line equivalent. You don't scare me demonic voice in my head.

other mathematical models

The pressure drag equation derived above is to me the most reasonable mathematical model of drag - especially aerodynamic drag. But as the demonic voice in my head said, it isn't always the easiest one to work with - especially for those just learning calculus (differential equations to be more precise). Those who know a lot of calculus just deal with it. Those who don't know any calculus just ignore it.

A simplified model of drag is one that assumes that drag is directly proportional to speed.. This sometimes is good enough. (Maybe we should call it the "good enough model of drag".) It is especially useful when teaching calculus students how to solve differential equations for the first time. I haven't found it to be all that applicable to real world situations, however. (We'll use b as the generic constant of proportionality from now on.)

A more general model of drag is one that is agnostic about higher powers (pun intended). This is good attitude to have when you are exploring drag experimentally. Don't assume you know anything about how drag varies with speed, just measure the two quantities and see what values work best for the power n and the constant of proportionality b.

Possibly the most general model is one that assumes a polynomial relationship. Drag might be related to speed in a way that is partially linear, partially quadratic, partially cubic, and partially described by higher order terms.

drag and power

While that's certainly true, it isn't of much use to us here on earth. If we lived in the vacuum of space, all we'd ever have to worry about was the energy needed to change our state from one speed to another. Here on earth, the atmosphere has another opinion. Whatever energy we add to a system to get it going, the atmosphere drags it away - all of it eventually. In order for a moving body to stay in motion on the Earth it not only has to get going, it has to actively work to keep going. This undeniable fact of life is why Newton's first law (the law of inertia) wasn't discovered until the 17th century.

To keep an object in motion in the presence of drag (aerodynamic or otherwise) requires an ongoing input of energy. Work must be done over some time. Power must be used. Recall the following chain of reasoning that starts from the definition of power as the rate at which work is done...

Replace the generic force variable with a generic power equation for drag...

Thus in general...

or more specifically, in the case of pressure drag...

Power expended against drag is the biggest impediment to moving freely for both bicycles and motorcycles. Humans can do sustained physical work like cycling at the rate of about a tenth of a horsepower. Motorcycles have engines that are on the order of 100 horsepower. (Sorry for the American units.) That makes a motorcycle about one thousand times more powerful than a human on a bicycle. As a result they can go about ten times faster, since 1000 = 10 ³. I've found through personal experience on all day bicycle rides that I typically cover ⅙ the distance that I would if I sat behind the wheel of a car all day.

Yes I realize that cars aren't motorcycles, but what we're really comparing here are wheeled vehicles powered by human muscle with those powered by internal combustion engines. Yes I realize that a 6 to 1 ratio is not exactly the same as 10 to 1, but what I'm doing here is a quick order of magnitude comparison. Your individual results may vary - but not significantly.

terminal velocity

It's much more than the name of a bad movie. It's something every student of aerodynamic drag should understand.

Imagine yourself as a parachute jumper; or better yet, imagine yourself as a BASE jumper. BASE is an acronym for building, antenna, span, e scarpment. Since none of these platforms is moving horizontally, none of these jumpers has any initial horizontal velocity. Not that it matters, but this reduces some of the complexity. Step off the platform and draw your free body diagram as you fall.

You start with no initial velocity, there is no aerodynamic drag, and you are effectively in free fall with an acceleration of 9.8 m/s ².

Now it gets complicated. There is an initial acceleration, therefore there is an increase in speed. With an increase in speed comes an increase in drag and a decrease in net force. This decrease in net force reduces acceleration. Speed is still increasing, just not quite as fast as it was initially.

Speed continues to increase, but so too does drag. As drag increases, acceleration decreases. Eventually one can imagine a state when the drag and weight forces are equal. You are in equilibrium. You continue moving, but you cease accelerating. You have reached your terminal velocity. Given the usual posture of skydivers, the type of clothes they normally wear, and the conditions of the air near the surface of the Earth; your typical skydiver has a terminal velocity of 55 m/s (200 km/h or 125 mph). The speed that you have in this state is the one you will always acquire if you are given enough time.

That is until the parachute opens. Opening the chute significantly increases your projected area, which cranks up the aerodynamic drag proportionally. The upward drag force now exceeds the downward pull of gravity. The net force and acceleration are directed upward. Note: this does not mean the skydiver is moving upward. Acceleration does not determine the direction of motion of an object, it determines the direction of the change in motion. When a parachute is just opened, the velocity is down and the acceleration is up. Your speed decreases as a result, which is the whole point behind the parachute.

Speed decreases, so drag decreases. Drag decreases, so the net force decreases. Eventually the net force is zero, you stop accelerating, and you reach a new terminal velocity - one that makes landing more comfortable, something like 6 m/s (22 km/h or 13 mph) or less.

Note that a terminal velocity is not necessarily a maximum value. It's a limit that can be approached from either direction. An object could start off slow and speed up to a terminal velocity that's a maximum (like a skydiver stepping off a BASE) or it could start off fast and slow down to a terminal velocity that's a minimum (like a skydiver who's just opened her parachute). "Terminal" is a fancy way to say "end". A terminal velocity is one that you end up with. For falling objects, this occurs when drag equals weight.

Terminal velocity applies to situations besides skydiving. Drive your car with the accelerator in a constant position and you'll eventually reach a terminal velocity. The forward driving force of the tires on the road will eventually equal the backward drag force of the air (and the rolling resistance of the tires, which is discussed somewhere else in this book). Note how I said "eventually". Terminal velocity is a speed things approach but never quite reach. Proof of this statement requires calculus and will be discussed in the practice problems of this section.

Terminal velocity can have any value - including zero. What happens to a ship in the ocean when the propeller stops turning? The forward thrust goes away and all that's left is the backward drag. The ship goes slower and slower and slower until it stops (stops relative to any current, that is). The ship will reach a terminal velocity of zero. For large container ships this may take minutes of time and kilometers of distance, but it will eventually happen. If you don't have the time or the space and you really want to stop a large seagoing vessel, you need to run the engines in reverse. In this case it's thrust that stops the ship, not drag.

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Aerodynamic Drag - The Physics Hypertextbook

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The force on an object that resists its motion through a fluid is called drag. When the fluid is a gas like air, it is called aerodynamic drag or air resistance. When the fluid is a liquid like water it is called hydrodynamic drag, but never "water resistance".

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Science | Classroom-Based Assessments | Junior Cycle for Teachers (JCT)

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Science | Classroom-Based Assessments | Junior Cycle for Teachers (JCT)

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Discussion

pressure drag

The force on an object that resists its motion through a fluid is called drag. When the fluid is a gas like air, it is called aerodynamic drag or air resistance. When the fluid is a liquid like water it is called hydrodynamic drag, but never "water resistance".

Fluids are characterized by their ability to flow. In somewhat technical language, a fluid is any material that can't resist a shear force for any appreciable length of time. This makes them hard to hold but easy to pour, stir, and spread. Fluids have no definite shape but take on the shape of their container. (We'll ignore surface tension for the time being. It's really only significant on the small scale - small like the size of a drop.) Fluids are polite in a sense. They yield their space relatively easily to other material things; at least when compared to solids. A fluid will get out of your way if you ask it. A solid has to be told to get out of the way with destructive force.

Fluids may not be solid, but they are most certainly material. The essential property of being material (in the classical sense) is to have both mass and volume. Material things resist changes in their velocity (this is what it means to have mass) and no two material things may occupy the same space at the same time (this is what it means to have volume). The portion of the drag force that is due to the inertia of the fluid - the resistance that it has to being pushed aside - is called the pressure drag (or form drag or profile drag). This is usually what someone is referring to when they talk about drag.

Recall Bernoulli's equation for the pressure in a fluid...

The first term on each side of the equation is the part of the pressure that comes from outside the fluid. Typically, this refers to atmospheric pressure weighing down on the surface of a liquid (not relevant right now). The second term is the gravitational contribution to pressure. This is what causes buoyancy (also not relevant right now). The third term is the kinetic or dynamic contribution to pressure - the part related to flow (very relevant right now). This will help us understand the origin of pressure drag.

Start with the definition of pressure as force per area. Solve it for force.

Replace the generic symbol F for force with the more specific symbol R for drag. (You could also use D if you wanted to.) Drop in Bernoulli's equation for the pressure in a moving fluid...

Rearrange things a bit and here you go...

Wait a minute. Where'd that extra symbol come from? Who put that C in there and why?

Let's run through all the symbols one at a time, explain their meaning and how they relate to pressure drag. In essence, let's take the equation apart and put it back together again.

Drag increases with the density of the fluid (ρ). More density means more mass, which means more inertia, which means more resistance to getting out of the way. The two quantities are directly proportional.
R ∝ ρ

Drag increases with area (A). Exactly what we mean by this is subject to debate. To me, and in the context of this model, area is the cross sectional area projected in the direction of motion. (I would further simplify this by calling it the projected area.) Take the cross section of the object in the direction of its motion. This is the area of the tube of fluid that must be cast aside to let the object pass. This is the most logical thing to call the area, but not everyone agrees with me. To some, the word "area" refers to the area of contact between the object and the fluid. This also makes sense, but not in the context I've described above. Surface area is not important when one is dealing with pressure drag, but it is important when dealing with viscous drag - drag caused by layers of the fluid sticking to the object and to one another. More surface area means more of the object is in contact with the fluid, which means more drag. Viscous drag is just as real as pressure drag, but I don't want to deal with it right now.

Drag increases with speed (v). I hope that this is self-evident. An object that is stationary with respect to the fluid will certainly not experience any drag force. Start moving and a resistive force will arise. Get moving faster and surely the resistive force will be greater. The hard part of this relationship lies in the detailed way speed affects drag. According to our sensible model derived from Bernoulli's sensible equation, drag should sensibly be proportional to the square of speed. In some situations, however, this may not be entirely correct. Drag is a complex phenomena. It cannot always be described with equations that are simple. My first guess would always be that drag is proportional to the square of speed since I understand and like the derviation I've presented, but I would not be surprised if (over some range of values) drag and speed were found to be directly proportional, proportional to some power other than 2, or related by some polynomial. Welcome to the world of empirical modeling - where relationships are determined by actual physical experiments rather than an ideology of pure theory. More on this subject later in this section.

Which brings us to our last factor...

Drag is influenced by other factors including shape, texture, viscosity (which results in viscous drag or skin friction), compressibility, lift (which causes induced drag), boundary layer separation, and so on. These factors can be dealt with separately in a more complete theory of drag (how tedious in one sense, but how necessary in another) or they can be piled into one monolithic fudge factor (oh yes, please) called the coefficient of drag (C).

Combining all these factors together yields a theoretically limited (but empirically reasonable) equation. Here it is again...

Simple, compact, wonderful. A nice equation to work with - or is it?

Well, yes and no.

1. Yes, but it works only as long as the range of conditions examined is "small". That is, no large variations in speed, viscosity, or crazy angles of attack. The way around this is to reduce the coefficient of drag to a variable rather than a constant. (I can live with this.) Say that C depends on some yet to be specified set of factors. It is totally acceptable to say that it varies with this that or the other quantity according to any set of rules determined by experiment.

2. No, since speed is squared. [Gasp!] Recall that speed is the derivative of distance with respect to time. Have you ever tried to solve a nonlinear differential equation? No? Well, welcome to hell. Wait, let me rephrase that - Welcome to Hell! [Ca-rack! Boom!] Ah ha ha ha ha haaaa! [Rumble] You fool! Just wait till you see what's in store for you when you try to solve the differential equations. The mathematics will consume you. [Ca-rack! Boom!] Ah ha ha ha ha haaaa! [Rumble].

Whew. What the hell was that all about? I might not know how to solve every kind of differential equation off the top of my head, but so what. I can always look for the solution in a book of standard mathematical tables or an on-line equivalent. You don't scare me demonic voice in my head.

other mathematical models

The pressure drag equation derived above is to me the most reasonable mathematical model of drag - especially aerodynamic drag. But as the demonic voice in my head said, it isn't always the easiest one to work with - especially for those just learning calculus (differential equations to be more precise). Those who know a lot of calculus just deal with it. Those who don't know any calculus just ignore it.

A simplified model of drag is one that assumes that drag is directly proportional to speed.. This sometimes is good enough. (Maybe we should call it the "good enough model of drag".) It is especially useful when teaching calculus students how to solve differential equations for the first time. I haven't found it to be all that applicable to real world situations, however. (We'll use b as the generic constant of proportionality from now on.)

A more general model of drag is one that is agnostic about higher powers (pun intended). This is good attitude to have when you are exploring drag experimentally. Don't assume you know anything about how drag varies with speed, just measure the two quantities and see what values work best for the power n and the constant of proportionality b.

Possibly the most general model is one that assumes a polynomial relationship. Drag might be related to speed in a way that is partially linear, partially quadratic, partially cubic, and partially described by higher order terms.

drag and power

While that's certainly true, it isn't of much use to us here on earth. If we lived in the vacuum of space, all we'd ever have to worry about was the energy needed to change our state from one speed to another. Here on earth, the atmosphere has another opinion. Whatever energy we add to a system to get it going, the atmosphere drags it away - all of it eventually. In order for a moving body to stay in motion on the Earth it not only has to get going, it has to actively work to keep going. This undeniable fact of life is why Newton's first law (the law of inertia) wasn't discovered until the 17th century.

To keep an object in motion in the presence of drag (aerodynamic or otherwise) requires an ongoing input of energy. Work must be done over some time. Power must be used. Recall the following chain of reasoning that starts from the definition of power as the rate at which work is done...

Replace the generic force variable with a generic power equation for drag...

Thus in general...

or more specifically, in the case of pressure drag...

Power expended against drag is the biggest impediment to moving freely for both bicycles and motorcycles. Humans can do sustained physical work like cycling at the rate of about a tenth of a horsepower. Motorcycles have engines that are on the order of 100 horsepower. (Sorry for the American units.) That makes a motorcycle about one thousand times more powerful than a human on a bicycle. As a result they can go about ten times faster, since 1000 = 10 ³. I've found through personal experience on all day bicycle rides that I typically cover ⅙ the distance that I would if I sat behind the wheel of a car all day.

Yes I realize that cars aren't motorcycles, but what we're really comparing here are wheeled vehicles powered by human muscle with those powered by internal combustion engines. Yes I realize that a 6 to 1 ratio is not exactly the same as 10 to 1, but what I'm doing here is a quick order of magnitude comparison. Your individual results may vary - but not significantly.

terminal velocity

It's much more than the name of a bad movie. It's something every student of aerodynamic drag should understand.

Imagine yourself as a parachute jumper; or better yet, imagine yourself as a BASE jumper. BASE is an acronym for building, antenna, span, e scarpment. Since none of these platforms is moving horizontally, none of these jumpers has any initial horizontal velocity. Not that it matters, but this reduces some of the complexity. Step off the platform and draw your free body diagram as you fall.

You start with no initial velocity, there is no aerodynamic drag, and you are effectively in free fall with an acceleration of 9.8 m/s ².

Now it gets complicated. There is an initial acceleration, therefore there is an increase in speed. With an increase in speed comes an increase in drag and a decrease in net force. This decrease in net force reduces acceleration. Speed is still increasing, just not quite as fast as it was initially.

Speed continues to increase, but so too does drag. As drag increases, acceleration decreases. Eventually one can imagine a state when the drag and weight forces are equal. You are in equilibrium. You continue moving, but you cease accelerating. You have reached your terminal velocity. Given the usual posture of skydivers, the type of clothes they normally wear, and the conditions of the air near the surface of the Earth; your typical skydiver has a terminal velocity of 55 m/s (200 km/h or 125 mph). The speed that you have in this state is the one you will always acquire if you are given enough time.

That is until the parachute opens. Opening the chute significantly increases your projected area, which cranks up the aerodynamic drag proportionally. The upward drag force now exceeds the downward pull of gravity. The net force and acceleration are directed upward. Note: this does not mean the skydiver is moving upward. Acceleration does not determine the direction of motion of an object, it determines the direction of the change in motion. When a parachute is just opened, the velocity is down and the acceleration is up. Your speed decreases as a result, which is the whole point behind the parachute.

Speed decreases, so drag decreases. Drag decreases, so the net force decreases. Eventually the net force is zero, you stop accelerating, and you reach a new terminal velocity - one that makes landing more comfortable, something like 6 m/s (22 km/h or 13 mph) or less.

Note that a terminal velocity is not necessarily a maximum value. It's a limit that can be approached from either direction. An object could start off slow and speed up to a terminal velocity that's a maximum (like a skydiver stepping off a BASE) or it could start off fast and slow down to a terminal velocity that's a minimum (like a skydiver who's just opened her parachute). "Terminal" is a fancy way to say "end". A terminal velocity is one that you end up with. For falling objects, this occurs when drag equals weight.

Terminal velocity applies to situations besides skydiving. Drive your car with the accelerator in a constant position and you'll eventually reach a terminal velocity. The forward driving force of the tires on the road will eventually equal the backward drag force of the air (and the rolling resistance of the tires, which is discussed somewhere else in this book). Note how I said "eventually". Terminal velocity is a speed things approach but never quite reach. Proof of this statement requires calculus and will be discussed in the practice problems of this section.

Terminal velocity can have any value - including zero. What happens to a ship in the ocean when the propeller stops turning? The forward thrust goes away and all that's left is the backward drag. The ship goes slower and slower and slower until it stops (stops relative to any current, that is). The ship will reach a terminal velocity of zero. For large container ships this may take minutes of time and kilometers of distance, but it will eventually happen. If you don't have the time or the space and you really want to stop a large seagoing vessel, you need to run the engines in reverse. In this case it's thrust that stops the ship, not drag.

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Aerodynamic Drag - The Physics Hypertextbook

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The force on an object that resists its motion through a fluid is called drag. When the fluid is a gas like air, it is called aerodynamic drag or air resistance. When the fluid is a liquid like water it is called hydrodynamic drag, but never "water resistance".

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Lab: Cupcakes and Drag Introduction To understand this experiment, one must first understand the effect of terminal velocity on Newton's Second Law (...

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Episode 209: Drag, air resistance, terminal velocity

Lesson for 16-19

This episode starts by considering the forces acting on a falling body in air, and moves on to related experimental work.

Lesson Summary

Discussion and demonstration: Falling bodies (10 minutes)

Student experiment: Falling cupcakes (20 minutes)

Student investigation/demonstration: Balls falling through fluids (20 minutes)

Discussion and demonstrations: Falling bodies

As an introduction to the lesson the following questions can be posed to the class. During the discussion, you will be able to introduce and define the terms upthrust, drag, air resistance, lift and terminal velocity.

Why does a piece of paper fall more slowly under gravity than a piece of chalk if the acceleration due to gravity is the same for all objects? (Demonstrate this.)

If an object is falling through the air with constant velocity, what can you say about the net force on the object?

How do the forces on an object vary as the object accelerates from rest?

The discussion should lead to the concept of an object reaching terminal velocity when the drag force has the same magnitude as the accelerating force. This can be shown clearly in a diagram such as this:

A useful exercise is to ask a student to sketch the velocity-time graph of a skydiver who accelerates to terminal velocity and then opens her parachute.She will decelerate to a new, lower terminal velocity. You can use a toy parachutist to illustrate this.

The concept of upthrust is most easily introduced using a toy helium balloon (see Balloons - Party in Yellow pages). It will accelerate as its upthrust is greater than its weight, but will reach an upward terminal velocity when upthrust = weight &plus; drag. If you have a high stairwell you can try taking measurements with students standing at different heights and timing the passage of the balloon - fun if you have the time.

An aeroplane is flying at constant velocity (and therefore at constant height). The net force on the aeroplane must be zero. How are the forces balanced? The forward thrust provided by the engines balances the backward drag of air resistance; the downward weight of the plane is balanced by the upward lift on the wings.

This lift is often explained by the Bernoulli effect - faster moving air over the top of the wing produces a pressure difference and an upward force. There are many easy demonstrations of this effect - the simplest being to blow between two sheets of paper causing them to stick together. However, you need to be a little careful with this explanation; remember that planes can fly upside down. The angle of the aerofoil is also important. You may prefer to avoid mentioning Bernoulli and simply say that the wing of an aircraft is shaped and angled to push air downwards; since there is a downward force of the wing on the air, there will be a corresponding upward force of the air on the wing (Newton's third law); this is the force we call lift.

Students will now be familiar with drag, thrust, lift, upthrust and weight.

Student experiment: Falling cupcakes

This works well if you make sure that the students are thinking about the questions posed. The activity itself is trivial but a lot of good physics can be brought out. It follows on very well from the introductory discussion and some students will find it useful to sketch force diagrams to help their answers. A written discussion of what the investigation shows could usefully be left for homework.

Demonstration: Balls falling through fluids

A more quantitative investigation of drag and terminal velocity is to observe the motion of bodies falling through a viscous medium. This is a rich area for student investigations but can be carried out as a demonstration by the teacher with students making measurements.

Start by dropping a ball bearing through the fluid. Can your students describe its motion? (At first it accelerates; acceleration gradually decreases to zero, when the ball has reached terminal velocity.)

The physics behind the demonstration is Stokes' Law and you can use this as an opportunity to discuss this or just remind students of a little kinematics as they analyse the motion of the falling ball bearing.
Episode 209-1: Falling cupcakes (Word, 30 KB) Episode 209-2: Ball bearings falling through a viscous medium (Word, 35 KB)

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Episode 209: Drag, air resistance, terminal velocity

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Lesson for 16-19 This episode starts by considering the forces acting on a falling body in air, and moves on to related experimental work. Lesson Summary Discussion and demonstration: Falling bodies (10 minutes) Student experiment: Falling cupcakes (20 minutes) Student investigation/demonstration: Balls falling through fluids (20 minutes) Discussion and demonstrations: Falling bodies As an introduction to the lesson the following questions can be posed to the class.

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Learning Objectives
By the end of the section, you will be able to:

Express the drag force mathematically

Describe applications of the drag force

Define terminal velocity

Determine an object's terminal velocity given its mass

Another interesting force in everyday life is the force of drag on an object when it is moving in a fluid (either a gas or a liquid). You feel the drag force when you move your hand through water. You might also feel it if you move your hand during a strong wind. The faster you move your hand, the harder it is to move. You feel a smaller drag force when you tilt your hand so only the side goes through the air-you have decreased the area of your hand that faces the direction of motion.

Drag Forces

Like friction, the drag force always opposes the motion of an object. Unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid. This functionality is complicated and depends upon the shape of the object, its size, its velocity, and the fluid it is in. For most large objects such as cyclists, cars, and baseballs not moving too slowly, the magnitude of the drag force is proportional to the square of the speed of the object. We can write this relationship mathematically as F D∝v 2. F D∝v 2. When taking into account other factors, this relationship becomes

where C is the drag coefficient, A is the area of the object facing the fluid, and ρ ρ is the density of the fluid. (Recall that density is mass per unit volume.) This equation can also be written in a more generalized fashion as F D=b v n, F D=b v n, where b is a constant equivalent to 0.5 C ρ A. 0.5 C ρ A. We have set the exponent n for these equations as 2 because when an object is moving at high velocity through air, the magnitude of the drag force is proportional to the square of the speed. As we shall see in Fluid Mechanics, for small particles moving at low speeds in a fluid, the exponent n is equal to 1.

Drag Force

Athletes as well as car designers seek to reduce the drag force to lower their race times ( Figure 6.29). Aerodynamic shaping of an automobile can reduce the drag force and thus increase a car's gas mileage.

The value of the drag coefficient C is determined empirically, usually with the use of a wind tunnel ( Figure 6.30).

The drag coefficient can depend upon velocity, but we assume that it is a constant here. Table 6.2 lists some typical drag coefficients for a variety of objects. Notice that the drag coefficient is a dimensionless quantity. At highway speeds, over 50% 50% of the power of a car is used to overcome air drag. The most fuel-efficient cruising speed is about 70-80 km/h (about 45-50 mi/h). For this reason, during the 1970s oil crisis in the United States, maximum speeds on highways were set at about 90 km/h (55 mi/h).

Object

C

Airfoil

0.05

Toyota Camry

0.28

Ford Focus

0.32

Honda Civic

0.36

Ferrari Testarossa

0.37

Dodge Ram Pickup

0.43

Sphere

0.45

Hummer H2 SUV

0.64

Skydiver (feet first)

0.70

Bicycle

0.90

Skydiver (horizontal)

1.0

Circular flat plate

1.12

Substantial research is under way in the sporting world to minimize drag. The dimples on golf balls are being redesigned, as are the clothes that athletes wear. Bicycle racers and some swimmers and runners wear full bodysuits. Australian Cathy Freeman wore a full body suit in the 2000 Sydney Olympics and won a gold medal in the 400-m race. Many swimmers in the 2008 Beijing Olympics wore (Speedo) body suits; it might have made a difference in breaking many world records ( Figure 6.31). Most elite swimmers (and cyclists) shave their body hair. Such innovations can have the effect of slicing away milliseconds in a race, sometimes making the difference between a gold and a silver medal. One consequence is that careful and precise guidelines must be continuously developed to maintain the integrity of the sport.

Terminal Velocity

Some interesting situations connected to Newton's second law occur when considering the effects of drag forces upon a moving object. For instance, consider a skydiver falling through air under the influence of gravity. The two forces acting on him are the force of gravity and the drag force (ignoring the small buoyant force). The downward force of gravity remains constant regardless of the velocity at which the person is moving. However, as the person's velocity increases, the magnitude of the drag force increases until the magnitude of the drag force is equal to the gravitational force, thus producing a net force of zero. A zero net force means that there is no acceleration, as shown by Newton's second law. At this point, the person's velocity remains constant and we say that the person has reached his terminal velocity (v T). (v T). Since is proportional to the speed squared, a heavier skydiver must go faster for to equal his weight. Let's see how this works out more quantitatively.

At the terminal velocity,

F net =m g−F D=m a=0. F net =m g−F D=m a=0.

Thus,

Using the equation for drag force, we have

Solving for the velocity, we obtain

Assume the density of air is ρ=1.21 kg/m 3. ρ=1.21 kg/m 3. A 75-kg skydiver descending head first has a cross-sectional area of approximately A=0.18 m 2 A=0.18 m 2 and a drag coefficient of approximately C=0.70 C=0.70 . We find that

v T=2(75 kg)(9.80 m/s 2)(1.21 kg/m 3)(0.70)(0.18 m 2) =98 m/s=350 km/h. v T=2(75 kg)(9.80 m/s 2)(1.21 kg/m 3)(0.70)(0.18 m 2) =98 m/s=350 km/h.

This means a skydiver with a mass of 75 kg achieves a terminal velocity of about 350 km/h while traveling in a headfirst position, minimizing the area and his drag. In a spread-eagle position, that terminal velocity may decrease to about 200 km/h as the area increases. This terminal velocity becomes much smaller after the parachute opens.

Example 6.17

Check Your Understanding 6.10

The size of the object that is falling through air presents another interesting application of air drag. If you fall from a 5-m-high branch of a tree, you will likely get hurt-possibly fracturing a bone. However, a small squirrel does this all the time, without getting hurt. You do not reach a terminal velocity in such a short distance, but the squirrel does.

The following interesting quote on animal size and terminal velocity is from a 1928 essay by a British biologist, J. B. S. Haldane, titled "On Being the Right Size."

"To the mouse and any smaller animal, [gravity] presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, and a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object. Divide an animal's length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistance to falling in the case of the small animal is relatively ten times greater than the driving force."

The above quadratic dependence of air drag upon velocity does not hold if the object is very small, is going very slow, or is in a denser medium than air. Then we find that the drag force is proportional just to the velocity. This relationship is given by Stokes' law.

Stokes' Law

Good examples of Stokes' law are provided by microorganisms, pollen, and dust particles. Because each of these objects is so small, we find that many of these objects travel unaided only at a constant (terminal) velocity. Terminal velocities for bacteria (size about 1 μm) 1 μm) can be about 2 μm/s. 2 μm/s. To move at a greater speed, many bacteria swim using flagella (organelles shaped like little tails) that are powered by little motors embedded in the cell.

Sediment in a lake can move at a greater terminal velocity (about 5 μm/s), 5 μm/s), so it can take days for it to reach the bottom of the lake after being deposited on the surface.

If we compare animals living on land with those in water, you can see how drag has influenced evolution. Fish, dolphins, and even massive whales are streamlined in shape to reduce drag forces. Birds are streamlined and migratory species that fly large distances often have particular features such as long necks. Flocks of birds fly in the shape of a spearhead as the flock forms a streamlined pattern ( Figure 6.32). In humans, one important example of streamlining is the shape of sperm, which need to be efficient in their use of energy.

The Calculus of Velocity-Dependent Frictional Forces

When a body slides across a surface, the frictional force on it is approximately constant and given by μ k N. μ k N. Unfortunately, the frictional force on a body moving through a liquid or a gas does not behave so simply. This drag force is generally a complicated function of the body's velocity. However, for a body moving in a straight line at moderate speeds through a liquid such as water, the frictional force can often be approximated by

where b is a constant whose value depends on the dimensions and shape of the body and the properties of the liquid, and v is the velocity of the body. Two situations for which the frictional force can be represented by this equation are a motorboat moving through water and a small object falling slowly through a liquid.

Let's consider the object falling through a liquid. The free-body diagram of this object with the positive direction downward is shown in Figure 6.33. Newton's second law in the vertical direction gives the differential equation

where we have written the acceleration as d v/d t. d v/d t. As v increases, the frictional force - bv increases until it matches mg. At this point, there is no acceleration and the velocity remains constant at the terminal velocity v T. v T. From the previous equation,

so

We can find the object's velocity by integrating the differential equation for v. First, we rearrange terms in this equation to obtain

d v g−(b/m)v=d t. d v g−(b/m)v=d t.

Assuming that integration of this equation yields

d v′g−(b/m)v′ =, d v′g−(b/m)v′ =,

or

−m b ln(g−b m v′)| 0 v=t′| 0 t, −m b ln(g−b m v′)| 0 v=t′| 0 t,

where v'and t' v'and t' are dummy variables of integration. With the limits given, we find

−m b[ln(g−b m v)−ln g]=t. −m b[ln(g−b m v)−ln g]=t.

Since ln A−ln B=ln(A/B), ln A−ln B=ln(A/B), and ln(A/B)=x implies e x=A/B, ln(A/B)=x implies e x=A/B, we obtain

and

Notice that as t→∞,v→m g/b=v T, t→∞,v→m g/b=v T, which is the terminal velocity.

The position at any time may be found by integrating the equation for v. With v=d y/d t, v=d y/d t,

Assuming

=(1−e−b t'/m) d t′, =(1−e−b t'/m) d t′,

which integrates to

y=t+m 2 g b 2 (−1). y=t+m 2 g b 2 (−1).

Example 6.18

Solution

1. With the motor stopped, the only horizontal force on the boat is f R=−b v, f R=−b v, so from Newton's second law, which we can write as Integrating this equation between the time zero when the velocity isv 0 v 0 and the time t when the velocity is v v , we have

d v′v′ =−b m d t′. d v′v′ =−b m d t′.

Thus, which, since ln A=x implies e x=A, ln A=x implies e x=A, we can write this as Now from the definition of velocity, so we have With the initial position zero, we have

d x'=v 0∫0 t e−b t'/m d t', d x'=v 0∫0 t e−b t'/m d t',

and

x=−m v 0 b e−b t'/m| 0 t=(1−). x=−m v 0 b e−b t'/m| 0 t=(1−).

As time increases, e−b t/m→0, e−b t/m→0, and the position of the boat approaches a limiting value Although this tells us that the boat takes an infinite amount of time to reachx max , x max , the boat effectively stops after a reasonable time. For example, at t=10 m/b, t=10 m/b, we have

v=v 0 e−10 ≃4.5×10 −5 v 0, v=v 0 e−10 ≃4.5×10 −5 v 0,

whereas we also have

x=(1−e−10 )≃0.99995. x=(1−e−10 )≃0.99995.

Therefore, the boat's velocity and position have essentially reached their final values.

2. With v 0=4.0 m/s v 0=4.0 m/s and v=1.0 m/s, v=1.0 m/s, we have 1.0 m/s=(4.0 m/s)e−(b/m)(10 s), 1.0 m/s=(4.0 m/s)e−(b/m)(10 s), so

ln 0.25=−ln 4.0=−b m(10 s), ln 0.25=−ln 4.0=−b m(10 s),

and

b m=1 10 ln 4.0 s -1 =0.14 s -1 . b m=1 10 ln 4.0 s -1 =0.14 s -1 .

Now the boat's limiting position is

==4.0 m/s 0.14 s −1 =29 m. ==4.0 m/s 0.14 s −1 =29 m.

Check Your Understanding 6.11

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Drag Force and Terminal Speed - University Physics Volume 1 - OpenStax

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Learning Objectives By the end of the section, you will be able to: Express the drag force mathematically Describe applications of the drag force Define terminal velocity Determine an object's terminal velocity given its mass Another interesting force in everyday life is the force of drag on an object when it is moving in a fluid (either a gas or a liquid).

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Air Resistance and Terminal Velocity - Complete Toolkit

Objectives

1. To explain the cause of air resistance.

2. To identify the variables that affect the amount of air resistance and to describe the manner in which those variables affect the amount of air resistance.

3. To describe the changes in speed, air resistance, net force and acceleration for an object falling under the influence of air resistance.

4. To explain why an object experiences a terminal velocity and why mass is an important factor affecting the terminal velocity value.

5. To compare and contrast free fall motion to the falling motion of an object experiencing air resistance.

Readings from The Physics Classroom Tutorial

1. The Physics Classroom Tutorial, Newton's Laws Chapter, Free Fall and Air Resistance

2. Multimedia Physics Studios, Newton's Laws Section, Skydiving

3. Multimedia Physics Studios, Newton's Laws Section, The Elephant and the Feather

Interactive Simulations

1. The Physics Interactives: Skydiving

The Skydiving Interactive simulates the motion of a falling skydiver. The mass of the skydiver, the parachute size, and the initial height can be varied. Learners view the size of the two individual forces, the velocity, and the height over the course of the falling motion. This simulation is part of The Physics Classroom's Physics Interactives collection. Like all the Interactives, it is written in HTML5 and compatible with Chromebooks, iPads, other tablets, and mobile phones. It is a perfect tool for the 1:1 classroom.

2. Free Fall Air Resistance Model

Powerful model for introductory physics. The blue ball falls under the influence of gravity alone; the red ball is influenced by air resistance. Students will have fun adjusting the air resistance, initial height, and velocity. Challenge them to find out gravitational constants on other planets and use those values to change the air resistance. (What would happen, for example, on a planet with low gravity but a dense atmosphere?) Written by respected EJS modeling author Andrew Duffy, professor of physics at Boston University. Available as a Java file only at this time.

Video and Animations

Labs and Investigations

1. The Physics Classroom, The Laboratory, Coffee Filter Skydiving Lab

Using a motion detector, students analyze the falling motion of a coffee filter and investigate how the velocity and the acceleration changes over time.

2. The Physics Classroom, The Laboratory, From a Feather to an Elephant

Using video analysis and footage of falling filters of varying mass, students determine the relationship between mass and terminal velocity. Mass is varied by nesting varying number of identical filters together.

3. The Physics Classroom, The Laboratory, Falling Body Spreadsheet Lab

Students utilize a spreadsheet to explore the motion of a falling body under the influence of air resistance. This open-ended exploration promotes experimental design and control of variables.

Link:The Laboratory

Demonstration Ideas

MIT Tech TV: Feather and Coin in a Vacuum

The free fall of coins and feathers are compared, first in a tube full of air, next with an evacuated vacuum pump. With air resistance, the feathers fall more slowly. In a vacuum, the objects fall at the same rate. This demo is also a Misconception Buster - See "Common Misconception #1" below.

Teachers: Of course, you can do this demo in class if you have a vacuum pump. Here's a link to lab procedures for performing the demo (from the UK's Institute of Physics) Lab Procedure: The Guinea and the Feather

Minds On Physics Internet Modules:

The Minds On Physics Internet Modules are a collection of interactive questioning modules that target a student's conceptual understanding. Each question is accompanied by detailed help that addresses the various components of the question.

1. Newton's Laws of Motion, Ass't NL10 - Free Fall Acceleration

2. Newton's Laws of Motion, Ass't NL11 - Air Resistance

Concept Building Exercises:

1. The Curriculum Corner, Newton's Laws, Air Resistance and Terminal Velocity

2. The Curriculum Corner, Newton's Laws, Skydiving

3. The Curriculum Corner, Newton's Laws, The Elephant and the Feather

Problem-Solving Exercises:

The Calculator Pad, Newton's Laws of Motion, Problems #17, 19, 29

Science Reasoning Activities:

Science Reasoning Center, Newton's Laws, Coffee Filter Physics Lab

Real Life Connections:

1. Air Resistance in Orbit

At the altitude of the International Space Station, there isn't much air. To stay in orbit, the ISS get regular "reboosts" from an Automated Transfer Vehicle. The boost causes the ISS to accelerate back up to proper orbit. But what happens to the astronauts when the ISS accelerates during a reboost? This 2-minute video shows you: if the astronauts don't anchor themselves, they will not move forward with the spacecraft - they appear to float backwards. But wait.....that's not all. Veteran physics blogger Rhett Allain has used this video as the basis of a really creative class activity to determine the ATV thrust based on video analysis of the astronauts' motion inside the ISS cabin. He takes you step-by-step through the entire process of problem set-up, quantitative analysis using kinematic equations, and help with data fitting.

2. NASA Glenn Research Center: Factors that Affect Drag

Want to extend the learning on air resistance, or provide extra challenges for students with high interest? NASA's Glenn Research Center has a nicely-organized set of introductory resources on aerodynamics. Here's the home page: NASA: Aerodynamics Index

Here are specific pages you might find useful for your unit on Air Resistance:

Common Misconception:

1. Heavier objects fall faster because they experience less air resistance.

Air resistance opposes the motion of a falling object. As an object falls, air resistance act upwards so as to reduce the net force. Heavy objects fall faster than lighter objects because the downward force (gravity) is bigger. A more massive object must accelerate to higher speeds in order to acquire an air resistance force that is large enough to balance the downward force. Thus, a heavier object will experience a greater amount of air resistance than a lighter object; it's just that it must have a higher terminal velocity value in order to obtain this larger air resistance value.

2. As an object approaches terminal velocity, both its velocity and acceleration decrease.

As a falling object approaches its terminal velocity, the net force is observed to decrease as the air resistance force increases. This decrease in net force is accompanied by a decreasing acceleration. These two statements are true. What often follows in a student's mind is the incorrect perception that the velocity is decreasing. This false belief exemplifies a student's tendency to confuse concepts of velocity and acceleration. The decrease in acceleration should be correctly understood to mean that the falling object is still picking up speed, just at a lower rate of acceleration. If the object were equipped with a speedometer, the needle would continue to show an increase in speed; however the rate at which the needle movesupward would decrease.

Elsewhere on the Web:

The Physics Classroom's Skydiving and Parachuting Gallery

This Flickr gallery of photos serves as either a suitable introduction to the topic of skydiving and parachuting or an enrichment activity for students interested in the topic. The gallery features 18 incredible photos from Flickr photographers, each annotated by a short description written in a light-hearted and everyday language. A student favorite.

Standards:

A. Next Generation Science Standards (NGSS)

Forces and Motion - High School - HS-PS2-6 Analyze data to support the claim that Newton's Second Law of Motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. (Strong Alignment)

Forces and Motion MS.PS2.A.ii The motion of an object is determined by the sum of forces acting on it; if the total force on the object is not zero, its motion will change. The greater the mass of the object, the greater the force needed to achieve the same change in motion. For any given object, a larger force causes a larger change in motion. (Strong Alignment)

Forces and Motion HS-PS1.A.i Newton's Second Law accurately predicts changes in the motion of macroscopic objects. (Strong Alignment)

High School: The functions and properties of natural and designed objects and systems can be inferred from their overall structure, the way their components are shaped and used, and the molecular substructures of its various materials.

High School: Change and rates of change can be quantified and modeled over very short or very long periods of time.

Nature of Science: Order & Consistency in Natural Systems

High School: Scientific knowledge is based on the assumption that natural laws operate today as they did in the past and they will continue to do so in the future.

Science and Engineering Practices
Practice #1: Analyzing and Interpreting Data

High School: Analyze data using tools, technologies, and/or models (e.g., computational, mathematical) in order to make valid and reliable scientific claims. (Strong alignment)

Practice #2: Developing and Using Models

High School: Use a model to provide mechanistic accounts of phenomena. (Strong alignment)

Practice #5: Using Mathematics and Computational Thinking

High School: Create or revise a computational model or simulation of a phenomenon, process, or system. (Strong alignment)

High School: Use mathematical representations of phenomena or design solutions to describe and/or support claims and/or explanations. (Strong alignment)

Practice #5: Using Mathematics and Computational Thinking

Create and/or revise a computational model or simulation of a phenomenon, designed device, process, or system.

Use mathematical and/or algorithmic representations of phenomena or design solutions to describe and/or support claims and explanations.

Apply techniques of algebra and functions to represent and solve scientific and engineering problems.

Apply ratios, rates, percentages, and unit conversions in the context of complicated measurement problems involving quantities with derived or compound units.

B. Common Core Standards for Mathematics - Grades 9-12

Standards for Mathematical Practice:

Reason abstractly and quantitatively

Model with mathematics

Functions - Interpreting Functions

F-IF.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

F-IF.6 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Linear, Quadratic, and Exponential Models

F-LE.1.b - Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

F-LE.5 - Interpret the parameters in a linear or exponential function in terms of a context.

C. Common Core Standards for English/Language Arts (ELA) - Grades 9-12

Note: Flesch-Kincaid Reading Ease Score for text in The Physics Classroom Tutorial-Newton's Second Law: 61.94, correlating to Grade 8.
Gunning-Fogg Index score: Grade 9

Reading Standards: Literacy in Science and Technical Subjects- Key Ideas & Details

RST11-12.3 - Follow precisely a complex multistep procedure when taking measurements or performing technical tasks; analyze the specific results based on explanations in the text.

RST.11-12.2 - Determine the central ideas or conclusions of a text; summarize complex concepts, processes, or information presented in a text by paraphrasing them in simpler but still accurate terms.

Reading Standards: Literacy in Science and Technical Subjects - Integration of Knowledge and Ideas

RST.11-12.7 - Integrate and evaluate multiple sources of information presented in diverse formats and media (e.g., quantitative data, video, multimedia) in order to address a question or solve a problem.

RST.11-12.9 - Synthesize information from a range of sources (e.g. texts, experiments, simulations) into a coherent understanding of a process, phenomenon, or concept.

Reading Standards: Literacy in Science and Technical Subjects - Range of Reading and Level of Text Complexity

RST.11-12.10 - By the end of Grade 10, read and comprehend science/technical texts in the Grade 9-10 text complexity band independently and proficiently

D. College Ready Physics Standards (Heller and Stewart)

Standard 3: Newton's Laws of Motion - Grades 9-12

Objective 3.2: Forces and Changes in Motion - Middle School

M.3.2.2 - Newton's second law of motion includes the following ideas:

a. When multiple forces are acting on an object, the change in motion of the object is determined by the sum of the forces (Newton's second law), which can be found using vector addition. The sum of the forces is not a real force caused by an interacting object; it is the single force that could replace the original multiple forces and cause the same change in motion.

M.3.2.3 - The forces acting on an object can be represented by arrows (vectors) drawn on an isolated picture of the object, called a force diagram. The direction of each arrow shows the direction of the push or pull. Forces are labeled: (type of interaction) push or pull of (interacting object) on the (object of interest). For example: "drag push of the wind on the sails of the boat".

Objective 3.2: Forces and Changes in Motion - High School

H.3.2.1 - Newton's second law of motion includes the following ideas:

a. The linear acceleration of an object is directly proportional to the vector sum of all the forces acting on the object and inversely proportional to the object's mass (a = F/m). The vector sum of all the forces (net force) is not a real force caused by an interaction with another object. The single force that could replace the original multiple forces and cause the same acceleration of the object is the vector sum of forces.

b. A special case of Newton's second law occurs when the vector sum of all the forces (net force) on an object is zero. In this case, there is no acceleration and the object remains at rest or maintains a constant speed and a constant direction of motion.

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standards-based physics lessons. air resistance

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