Air Resistance

Sidarth Rajan Fall 2020

Main Idea

When an object moves through a fluid (gas or liquid), forms of friction called air resistance, viscosity, and drag oppose the direction of motion of the object. Air resistance is one of the most common types of these oppositional forces. Its effects must be taken into account with any form of flight (space ships, airplanes, rockets, missiles). The drag force is a means of energy dissipation, where energy is converted into a non useful form, thermal energy.

The amount of air resistance is related to quite a few factors, such as cross sectional area, relative speed, and density.

Mathematical Model

A typical drag force can be described by:

[math]\displaystyle{ F_{D} = \frac{1}{2}\rho v^2 A C_{D} }[/math], where

[math]\displaystyle{ \rho = }[/math] the density of the fluid being passed through

[math]\displaystyle{ v = }[/math] the speed of the object relative to the fluid

[math]\displaystyle{ A }[/math] is the cross-sectional area of the object

[math]\displaystyle{ C_{D} = }[/math] a dimensionless drag coefficient

Cross Sectional Area

We have encountered the term of 'cross-sectional area' in many cases: for example, when you need to calculate the flux of a liquid flowing through a pipe, you need the cross-sectional area to determine the volume of the liquid flowing through. However, the 'cross-sectional area' here is different. We'd better see it as the 'frontal area': the area facing the direction of the object's movement.

It is easy to understand: we can relate this idea to the choice of the system and moving objects. For example, when a plane is flying forward with a speed [math]\displaystyle{ v }[/math] in the [math]\displaystyle{ +x }[/math] direction and the air is static, only the surface of the plane pointing in the [math]\displaystyle{ +x }[/math] direction is subject to the air resistance, since most air particles are colliding with that area with relatively high speed in the opposite direction.

Drag Coefficient

The drag coefficient is a dimensionless number that engineers use to model all of the complex dependencies of shape and flow conditions on motion resistance (drag). It is usually determined by experiment, since the factors are very complex. The drag coefficient is a function of several parameters like shape of the body, Reynolds Number for the flow, Froude number, Mach Number and Roughness of the Surface. In short, different materials, substrates, and solvents have different drag coefficients.

Computational Model

Insert Computational Model here

Examples

Simple

You are standing at the top of a [math]\displaystyle{ 20 }[/math] meter high building, and throw a ball with an initial speed of [math]\displaystyle{ v_{0} = 10 \frac{m}{s} }[/math] in the [math]\displaystyle{ +x }[/math] direction.

a) If you neglect air resistance, where would you expect the ball to hit on the flat surface below?

Without air resistance, the acceleration of the ball will be about a constant [math]\displaystyle{ g = 9.81 \ \frac{m}{s^2} }[/math]

We can use the kinematic equations under constant acceleration:

[math]\displaystyle{ v_{f} = v_{0} + at \ \ \mathbf{\text{(1)}} \quad \And \quad \Delta x = v_{0}t + \frac{1}{2}at^2 \ \ \mathbf{\text{(2)}} \quad \And \quad {v_f}^2 = {v_0}^2 + 2a \Delta x \ \ \mathbf{\text{(3)}} }[/math]

We will break this into two parts:

The y-direction (vertical)
The x-direction (horizontal)

The y-direction:

From the problem statement, we are told the ball is thrown horizontally, so there is no initial velocity along the y-direction:

[math]\displaystyle{ v_{y_{0}} = 0 }[/math]

Also, the only acceleration affecting the system, the acceleration due to gravity is along the negative y-direction:

[math]\displaystyle{ a_{y} = -g = -9.81 }[/math]

Also, we are told the initial height:

[math]\displaystyle{ y_{0} = 20 }[/math]

Using equation 3, we can find the final velocity along the y-direction of the ball:

[math]\displaystyle{ v_{y_{f}} = - \sqrt{{v_{y_{0}}}^2 + 2a_{y} \Delta y} = -\sqrt{0 + 2 \times (-9.81) \times (0 - 20)} = -\sqrt{392.4} = -19.81 \ \frac{m}{s} }[/math]

Now, we can plug this final velocity into 1 to find the time of flight for the ball:

[math]\displaystyle{ v_{y_{f}} = v_{y_{0}} + a_{y}t }[/math]

[math]\displaystyle{ t = \frac{\Delta v_{y}}{a_{y}} = \frac{-19.81}{-9.81} = 2.02 \ s }[/math]

Knowing the time of flight will allow us to solve for the distance traveled horizontally.

The x-direction:

We know the initial velocity in this direction:

[math]\displaystyle{ v_{x_{0}} = 10 }[/math]

Also, there is no acceleration along the x-direction:

[math]\displaystyle{ a_{x} = 0 }[/math]

Using equation 2, we can find how far the ball goes horizontally:

[math]\displaystyle{ \Delta x = v_{x_{0}}t + \frac{1}{2}a_{x}t^2 = v_{x_{0}} t = 10 \times 2.02 = 20.2 \ m }[/math]

Therefore, the ball will land a distance of 20.2 meters away horizontally and 20 meters below the roof of the building.

b) Do you think your prediction without air resistance is too large or too small? Explain.

If air resistance was taken into consideration, it would impede the motion of the ball, slowing it down. Thus, the previously predicted distance would be further than the ball would really go.

Middling

John is going sky diving for the first time. His mass is 70 kg and his terminal speed is 38 m/s.

a) What is the magnitude of the drag force on John?

Since John has reached terminal speed, he is not accelerating. Equivalently, this means his acceleration is [math]\displaystyle{ 0 \ \frac{m}{s^2} }[/math], his change in momentum is [math]\displaystyle{ 0 \ \frac{kg \cdot m}{s} }[/math], and the net force on him is [math]\displaystyle{ 0 \ \text{N} }[/math]. Thus, we know the two forces acting on him, the gravitational force and the drag force, are balanced:

[math]\displaystyle{ F_{net} = F_{D} - F_{g} = 0 }[/math]

[math]\displaystyle{ F_{D} = F_{g} }[/math]

Since John is close to the surface of Earth, we will use [math]\displaystyle{ mg }[/math] as the gravitational force:

[math]\displaystyle{ F_{D} = mg = 70 \times 9.81 = 686.7 \ \text{N} }[/math]

Therefore, the drag force on John is about [math]\displaystyle{ 686.7 }[/math] Newtons pointing away from the Earth.

b) What is an expression for the value of the cross-sectional area of John?

Using our drag force equation and setting it equal to the gravitational force, we see:

[math]\displaystyle{ F_{D} = \frac{1}{2}\rho v^2 A C_{D} }[/math]

[math]\displaystyle{ \frac{1}{2}\rho v^2 A C_{D} = mg }[/math]

Solving this for [math]\displaystyle{ A }[/math] gives:

[math]\displaystyle{ A = \frac{2mg}{\rho v^2 C_{D}} }[/math]

If we knew the density of the air and the drag coefficient, we could give a value for the cross-sectional are of John.

Difficult

Sarah is doing an air resistance experiment in class. The experiment requires Sarah to drop a coffee filter from a height of 2 meters. Let's say that the mass of the coffee filter was 2.0 grams, and it reached the ground with a speed of 1.0 m/s.

a) How much energy did the air gain as the coffee filter floated down?

This is a conservation of energy problem. We can describe the coffee filter as the system and the air as the surroundings to a good approximation. Thus, by the Energy Principle:

[math]\displaystyle{ \Delta E_{c} + \Delta E_{a} = 0 }[/math]

[math]\displaystyle{ \Delta E_{c} = - \Delta E_{a} }[/math]

The initial energy of the coffee filter can be described as a combination of the initial Thermal, Gravitational Potential, and Kinetic energies:

[math]\displaystyle{ E_{c_{0}} = U_{c_{0}} + K_{c_{0}} + Q_{c_{0}} }[/math]

The initial kinetic energy will be [math]\displaystyle{ 0 }[/math] since the coffee is dropped from rest:

[math]\displaystyle{ E_{c_{0}} = U_{c_{0}} + Q_{c_{0}} }[/math]

The final energy of the coffee filter can be described as a combination of the final Thermal, Gravitational Potential, and Kinetic energies:

[math]\displaystyle{ E_{c_{f}} = U_{c_{f}} + K_{c_{f}} + Q_{c_{f}} }[/math]

Subtracting the final energy from the initial energy will give us the change in energy of the coffee filter:

[math]\displaystyle{ \Delta E_{c} = E_{c_{f}} - E_{c_{0}} = (U_{c_{f}} + K_{c_{f}} + Q_{c_{f}}) - (U_{c_{0}} + Q_{c_{0}}) = (U_{c_{f}} - U_{c_{0}}) + K_{c_{f}} + (Q_{c_{f}} - Q_{c_{0}}) = \Delta U_{c} + K_{c_{f}} + \Delta Q_{c} }[/math]

Using common expressions for Gravitational Potential energy and Kinetic energy, we see:

[math]\displaystyle{ \Delta E_{c} = mg(h_{f} - h_{0}) + \left(\frac{1}{2}\right)m{v_f}^2 + \Delta Q_{c} }[/math]

We will assume the change in Thermal energy is negligible:

[math]\displaystyle{ \Delta E_{c} = mg(h_{f} - h_{0}) + \left(\frac{1}{2}\right)m{v_f}^2 }[/math]

We have all the needed values for this:

[math]\displaystyle{ \Delta E_{c} = 0.002 \times 9.81 \times (0 - 2) + \left(\frac{1}{2}\right) \times 0.002 \times (1.0)^2 = -0.03924 + 0.001 = -0.03824 \ J }[/math]

Using:

[math]\displaystyle{ \Delta E_{c} = - \Delta E_{a} }[/math]

We see:

[math]\displaystyle{ - \Delta E_{c} = \Delta E_{a} }[/math]

[math]\displaystyle{ \Delta E_{a} = -(-0.03824) = 0.03824 \ J }[/math]

We see the coffee filter lost [math]\displaystyle{ 0.03824 \ J }[/math] of energy, and the air in contact with the coffee filter gained it.

Connectedness

How is this topic connected to something that you are interested in?

As an adventurous person, I have always been interested in skydiving. Air resistance is a huge factor in skydiving, as it allows you to reach the ground safely just with a parachute. There are many factors in releasing a parachute to have the safest possible landing. When you release your parachute and how to control your parachute are very important in having a safe landing. Also, there is obviously a lot of air resistance in a parachute because of the large cross sectional area.

Another topic that air resistance plays a factor in is sports. A specific sport that air resistance impacts is tennis. Spin is very important in tennis, because it allows you to control where the ball lands. If there is topspin on a ball, the air resistance is less allowing the ball to come down faster. If there is backspin, the ball stays in the air longer, being controlled by the air.

Moreover, the simulation of motions in animations are also very interesting. If you ignore the air resistance of the motion, it will be absolutely very fake. Amazingly people have the sense to tell which scene is more actual and authentic. When you make an animation of snow flickers floating in the sky, it is important for you to precisely take air resistance forces into count.

How is it connected to your major?

I am a Biomolecular Engineering major. Although my major seems to have nothing related to air resistance, the topic can be generalized to many other resistance forces. For example, we can apply the resistance force equation to water, and simulate a diver's motion to find the best fitted materials for diving suits. Maybe it is also possible for us to develop some important first-aid machines or devices that can be easily carried through any kind of surrounding substances.

Is there an interesting industrial application?

Air resistance has a lot of application in speed sports. Also, air resistance plays a big factor in skydiving and anything with a parachute. Lastly, the aircraft industry factors in air resistance into all of their products, as this force is very important in takeoff, landing, and flight.

History

Aristotle was the first to write about air resistance in the 4th century BC. In the 15th century, Leonardo da Vinci published the Codex Leicester, in which he rejected Aristotle's theory and attempted to prove that the only effect of air on a thrown object was to resist its motion. The first equation for air resistance was:

[math]\displaystyle{ F_{D} = \rho S V^2 \text{sin}^2(\theta) }[/math]

This equation overestimates drag in most cases, and was often used in the 19th century to argue the impossibility of human flight.

Louis Charles Breguet's paper in 1922 began efforts to reduce drag by streamlining. A further major call for streamlining was made by Sir Melvill Jones who provided the theoretical concepts to demonstrate emphatically the importance of streamlining in aircraft design. The aspect of Jones’s paper that most shocked the designers of the time was his plot of the horse power required versus velocity, for an actual and an ideal plane.

Chabay, Ruth, and Bruce Sherwood. "Internal Energy." Matters and Interactions. 4th ed. Vol. 1. Wiley, 2015. Print.
More information can be found on drag[2] and aerodynamics[3]

Air Resistance

Contents