Express mathematically the drag force.Discuss the applications of traction force.Define terminal velocity.Determine the terminal velocity provided mass.

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Another exciting force in everyday life is the pressure of drag on an object when that is moving in a liquid (either a gas or a liquid). You feel the drag force when you move your hand v water. Girlfriend might additionally feel the if you move your hand during a strong wind. The quicker you relocate your hand, the harder it is come move. You feeling a smaller drag pressure when you tilt her hand so just the side goes through the air—you have diminished the area of your hand that encounters the direction of motion. Like friction, the traction force constantly opposes the movement of an object. Unlike basic friction, the drag force is proportional to some role of the velocity of the object in the fluid. This usability is facility and relies upon the form of the object, that is size, the velocity, and the fluid it is in. Because that most huge objects such as bicyclists, cars, and baseballs not moving too slowly, the magnitude of the drag force FD is found to be proportional come the square the the rate of the object. We deserve to write this partnership mathematically as F_ extDproptov^2\. When taking right into account various other factors, this relationship becomes F_ extD=frac12 extC hoAv^2\, where C is the drag coefficient, A is the area the the object dealing with the fluid, and ρ is the density of the fluid. (Recall that density is mass every unit volume.) This equation can also be created in a more generalized fashion together FD = bv2, whereby b is a consistent equivalent to 0.5CρA. Us have set the exponent n because that these equations together 2 because, when an object is moving at high velocity through air, the size of the drag force is proportional come the square that the speed. Together we shall view in a few pages on liquid dynamics, for small particles relocating at short speeds in a fluid, the exponent n is same to 1.

Drag Force

Drag force FD is discovered to it is in proportional to the square of the rate of the object. Mathematically

F_ extDproptov^2\

F_ extD=frac12 extC hoAv^2\,

where C is the drag coefficient, A is the area the the object encountering the fluid, and ρ is the density of the fluid.

Athletes and car developers seek to reduce the drag force to lower their gyeongju times. (See number 1). “Aerodynamic” shaping that an automobile can mitigate the drag force and so boost a car’s gas mileage.

Figure 1. From gyeongju cars to bobsled racers, aerodynamic shaping is an important to achieving peak speeds. Bobsleds room designed because that speed. They room shaped choose a bullet with tapered fins. (credit: U.S. Army, via Wikimedia Commons)

The value of the traction coefficient, C , is identified empirically, usually v the use of a wind tunnel. (See figure 2).

Figure 2. NASA researchers test a model airplane in a wind tunnel. (credit: NASA/Ames)

The traction coefficient can depend upon velocity, yet we will certainly assume the it is a continuous here. Table 1 lists some typical drag coefficients for a selection of objects. An alert that the traction coefficient is a dimensionless quantity. In ~ highway speeds, end 50% of the power of a vehicle is offered to overcome air drag. The most fuel-efficient cruising rate is about 70–80 km/h (about 45–50 mi/h). For this reason, throughout the 1970s oil dilemm in the united States, maximum speeds on highways were collection at about 90 km/h (55 mi/h).

Table 1. Drag Coefficient Values usual values of traction coefficient C.OBJECTC
Toyota Camry0.28
Ford Focus0.32
Honda Civic0.36
Ferrari Testarossa0.37
Dodge lamb pickup0.43
Hummer H2 SUV0.64
Skydiver (feet first)0.70
Skydiver (horizontal)1.0
Circular level plate1.12

Figure 3. Human body suits, such together this LZR Racer Suit, have been attributed with many people records after ~ their relax in 2008. Smoother “skin” and much more compression pressures on a swimmer’s body provide at least 10% less drag. (credit: NASA/Kathy Barnstorff)

Substantial study is under way in the sporting civilization to minimize drag. The dimples ~ above golf balls space being redesigned as room the clothes that athletes wear. Bicycle racers and also some swimmers and runners wear full bodysuits. Australian Cathy Freeman wore a complete body suit in the 2000 Sydney Olympics, and also won the yellow medal for the 400 m race. Plenty of swimmers in the 2008 Beijing Olympics wore (Speedo) body suits; it could have made a difference in breaking many people records (See figure 3). Most elite swimmers (and cyclists) cut their body hair. Together innovations deserve to have the result of slicing away milliseconds in a race, occasionally making the difference between a gold and a silver- medal. One consequence is that mindful and an exact guidelines need to be continuously occurred to preserve the truth of the sport.

Some exciting situations associated to Newton’s second law occur when considering the impacts of drag forces upon a relocating object. Because that instance, think about a skydiver falling v air under the influence of gravity. The two forces acting ~ above him space the pressure of gravity and also the drag force (ignoring the buoyant force). The downward force of gravity remains constant regardless of the velocity in ~ which the person is moving. However, together the who velocity increases, the size of the drag force increases till the magnitude of the drag force is equal to the gravitational force, thus developing a net force of zero. A zero network force way that there is no acceleration, as provided by Newton’s 2nd law. In ~ this point, the person’s velocity remains constant and us say that the person has actually reached his terminal velocity (vt). Due to the fact that FD is proportional come the speed, a heavier skydiver need to go faster for FD to same his weight. Stop see exactly how this works out much more quantitatively.

At the terminal velocity, F_ extnet=mg-F_ extD=ma=0\. Thus, mg=F_ extD\. Using the equation for drag force, we have mg=frac12 hoCAv^2\. Solving for the velocity, us obtain v=sqrtfrac2mg hoCA\.

Assume the thickness of waiting is ρ = 1.21 kg/m3. A 75-kg skydiver diminish head very first will have an area around A = 0.18 m2 and a traction coefficient of about C=0.70. We uncover that

eginarraylllv& =& sqrtfrac2left( ext75 kg ight)left(9 ext.80 m ext/s^2 ight)left(1 ext. ext21 kg ext/m^3 ight)left(0 ext. ext70 ight)left( ext0.18 extm^2 ight)\ & =& ext98 m/s\ & =& ext350 km/h ext.endarray\

This means a skydiver through a fixed of 75 kg achieves a maximum terminal velocity of around 350 km/h while travel in a pike (head first) position, minimizing the area and also his drag. In a spread-eagle position, that terminal velocity might decrease to about 200 km/h as the area increases. This terminal velocity becomes much smaller ~ the parachute opens.

Take-Home Experiment

This interesting task examines the effect of weight upon terminal velocity. Gather with each other some nested coffee filters. Leaving lock in their initial shape, measure the moment it takes for one, two, three, four, and also five nested filter to fall to the floor native the same elevation (roughly 2 m). (Note that, as result of the method the filters are nested, drag is continuous and just mass varies.) They obtain terminal velocity rather quickly, so uncover this velocity as a function of mass. Plot the terminal velocity v matches mass. Also plot v2 versus mass. I beg your pardon of these relationships is an ext linear? What have the right to you break up from this graphs?

Example 1. A Terminal Velocity

Find the terminal velocity of one 85-kg skydiver fallout’s in a spread-eagle position.


At terminal velocity, Fnet = 0. Therefore the drag force on the skydiver must equal the pressure of gravity (the person’s weight). Using the equation of drag force, we find mg=frac12 hoCAv^2\.

Thus the terminal velocity vt deserve to be created as v_ exttsqrtfrac2mg hoCA\.


All quantities are known other than the who projected area. This is an adult (82 kg) falling spread eagle. We can estimate the frontal area as A = (2 m)(0.35 m) = 0.70 m2.

Using ours equation for v, we uncover that

eginarraylllv_ extt& =& sqrtfrac2left( ext85 extkg ight)left(9.80 extm/s^2 ight)left(1.21 extkg/m^3 ight)left(1.0 ight)left(0.70 extm^2 ight)\ & =& ext44 m/s.endarray\


This result is consistent with the worth for vt discussed earlier. The 75-kg skydiver walk feet very first had a v = 98 m/s. He sweet less yet had a smaller sized frontal area and so a smaller drag due to the air.

The dimension of the object that is falling through air presents an additional interesting application of air drag. If you fall from a 5-m high branch that a tree, you will likely acquire hurt—possibly fracturing a bone. However, a small squirrel walk this all the time, without obtaining 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 native a 1928 essay through a british biologist, J.B.S. Haldane, titled “On gift the best Size.”

To the mouse and also any smaller animal, presents virtually no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on showing up at the bottom, it gets a slim shock and walks away, noted that the soil is fairly soft. A rat is killed, a man is broken, and also a equine splashes. Because that the resistance gift to activity by the wait is proportional to the surface of the relocating object. Division an animal’s length, breadth, and height each by ten; its weight is lessened to a thousandth, yet its surface just to a hundredth. So the resistance to falling in the case of the tiny animal is relatively ten times better than the control force.

The above quadratic dependency of air drag upon velocity walk not organize if the thing is an extremely small, is going really slow, or is in a denser medium than air. Climate we find that the drag force is proportional simply to the velocity. This partnership is provided by Stokes’ law, which claims that Fs = 6πrηv, where r is the radius of the object, η is the viscosity of the fluid, and also v is the object’s velocity.

Stokes’ Law

Fs = 6πrηv, where r is the radius that the object, η is the viscosity the the fluid, and v is the object’s velocity.

Figure 4. Geese fly in a V formation during their long migratory travels. This form reduces drag and energy intake for separation, personal, instance birds, and also also allows them a better way to communicate. (credit: Julo, Wikimedia Commons)

Good examples of this legislation are noted by microorganisms, pollen, and also dust particles. Because each of this objects is for this reason small, we uncover that many of these objects travel unaided just at a consistent (terminal) velocity. Terminal velocities because that bacteria (size about 1 μm) have the right to be about 2 μm/s. To relocate at a better speed, many bacteria swim using flagella (organelles shame like tiny tails) that room powered by small motors installed in the cell. Sediment in a lake deserve to move at a better terminal velocity (about 5μ m/s), so it deserve to take work to with the bottom of the lake after gift deposited top top the surface.

If us compare pets living ~ above land v those in water, you have the right to see how drag has influenced evolution. Fishes, dolphins, and even substantial whales are systematized in form to minimize drag forces. Birds space streamlined and migratory types that fly huge distances often have specific features such as lengthy necks. Flocks of birds fly in the form of a spear head together the flock develops a streamlined pattern (see figure 4). In humans, one vital example of streamlining is the shape of sperm, which have to be efficient in their use of energy.

Galileo’s Experiment

Galileo is said to have dropped two objects of various masses native the Tower of Pisa. He measured exactly how long it took each to with the ground. Due to the fact that stopwatches weren’t conveniently available, just how do girlfriend think he measured their loss time? If the objects to be the same size, but with various masses, what perform you think the should have actually observed? would certainly this an outcome be different if done on the Moon?

PhET Explorations: Masses & Springs

A realistic mass and also spring laboratory. Cave masses native springs and adjust the feather stiffness and damping. You can even slow time. Transport the lab to various planets. A chart reflects the kinetic, potential, and also thermal power for each spring.


Click to operation the simulation.

Section Summary

Drag forces acting on an item moving in a fluid oppose the motion. For bigger objects (such as a baseball) moving at a velocity v in air, the drag force is provided by F_ extD=frac12 extC hoAv^2\, where C is the traction coefficient (typical worths are given in Table 1), A is the area of the object dealing with the fluid, and ho\ is the fluid density.For little objects (such together a bacterium) moving in a denser medium (such together water), the drag pressure is offered by Stokes’ law, F_ exts=6pietarv\, where r is the radius of the object, η is the fluid viscosity, and also v is the object’s velocity.

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Conceptual Questions

Athletes such as swimmers and also bicyclists wear human body suits in competition. Build a list of pros and cons of together suits.Two expressions were supplied for the drag pressure experienced by a moving object in a liquid. One relied on the speed, when the other was proportional to the square of the speed. In which species of movement would each of this expressions be more applicable 보다 the various other one?As dare travel, oil and also gasoline leaks ~ above the road surface. If a irradiate rain falls, what does this perform to the control of the car? does a hefty rain make any type of difference?Why can a squirrel jump from a tree branch to the ground and run far undamaged, when a human might break a bone in such a fall?

Problems & Exercises

The terminal velocity the a human falling in air counts upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meter per second and kilometers every hour) of an 80.0-kg skydiver fallout’s in a pike (headfirst) position with a surface ar area the 0.140 m2.A 60-kg and a 90-kg skydiver run from an plane at one altitude of 6000 m, both falling in the pike position. Do some presumption on your frontal areas and calculate their terminal velocities. How long will certainly it take for each skydiver to reach the soil (assuming the time to reach terminal velocity is small)? i think all worths are specific to three far-reaching digits.A 560-g squirrel through a surface ar area that 930 cm2 falls from a 5.0-m tree come the ground. Calculation its terminal velocity. (Use a traction coefficient for a horizontal skydiver.) What will certainly be the velocity of a 56-kg person hitting the ground, suspect no drag contribution in such a brief distance?To keep a consistent speed, the force listed by a car’s engine need to equal the drag pressure plus the pressure of friction the the road (the roll resistance). (a) What space the magnitudes that drag forces at 70 km/h and also 100 km/h for a Toyota Camry? (Drag area is 0.70 m2) (b)What is the magnitude of drag pressure at 70 km/h and 100 km/h for a Hummer H2? (Drag area is 2.44 m2) i think all values are accurate to three significant digits.By what variable does the drag pressure on a car increase together it goes from 65 come 110 km/h?Calculate the rate a spherical rain fall would attain falling from 5.00 km (a) in the absence of air drag (b) through air drag. Take it the size across of the fall to it is in 4 mm, the thickness to be 1.00 × 103 kg/m3, and also the surface area to be πr2.Using Stokes’ law, verify the the units for viscosity are kilograms per meter per second.Find the terminal velocity the a spherical bacterium (diameter 2.00 μ m) falling in water. Friend will an initial need to note that the drag force is equal to the weight at terminal velocity. Take it the density of the bacter to it is in 1.10 × 103 kg/m3.Stokes’ law explains sedimentation of corpuscle in liquids and also can be offered to measure up viscosity. Corpuscle in liquids accomplish terminal velocity quickly. One can measure the time it takes because that a bit to autumn a specific distance and also then usage Stokes’ regulation to calculation the viscosity of the liquid. Expect a steel round bearing (density 7.8 × 103 kg/m3, diameter 3.0 mm) is reduce in a container of motor oil. It takes 12 s to fall a distance of 0.60 m. Calculate the viscosity that the oil.


drag force: FD, uncovered to it is in proportional to the square the the speed of the object; mathematically F_ extDpropto v^ ext2\, F_ extD=frac12C hoAv^2\, where C is the traction coefficient, A is the area of the object encountering the fluid, and ho is the density of the fluid

Stokes’ law: F_s=6piretav\ , where is the radius of the object, η is the viscosity the the fluid, and v is the object’s velocity

Selected solutions to troubles & Exercises

1. 115 m/s; 414 km/hr

3. 25 m/s; 9.9 m/s

5. 2.9

7. left=fracleftleftleft=frac extkgcdot extm/s^2 extmcdot extm/s=frac extkg extmcdot exts\