## Aerodynamic Drag

Practice problem 1.

- Determine the drag coefficient of a 75 kg skydiver with a projected area of 0.33 m 2 and a terminal velocity of 60 m/s.
- By how much would the skydiver need to reduce her project area so as to double her terminal velocity? How would she accomplish this?

Terminal velocity for a falling object occurs when the drag on the object equals its weight.

Solve for projected area, substitute values, and compute. (The density of air is in this book somewhere .)

This agrees with the range of values stated in the table on the discussion page of this topic.

We still start with the principle that drag equals weight.

But this time we'll solve for the terminal velocity instead of the drag coefficient.

Don't plug in any numbers, just look at the way terminal velocity is related to projected area. Projected area is in the denominator, under a radical sign. That means terminal velocity is inversely proportional to the square root of projected area. That means the skydiver would have to reduce her projected area to one-quarter of its original value.

The skydiver can do this by changing her orientation from horizontal to vertical — from spread eagle to head first.

## practice problem 2

Table 1 time (s) distance (m) 1 00 4.5472 2 0 17.5392 3 0 38.0016 4 0 64.96 00 5 0 97.44 00 6 134.4672 7 175.0672 8 218.2656 9 263.088 0 10 308.56 00 11 353.7072 Voici des Tables faites sur cette hypothèse, par lesquelles on connoîtra combien une balle de plomb de six lignes de diamètre passera de pieds en chaque seconde en descendant; combien elle en passera dans tel nombre de secondes qu'on voudra choisir; quand elle cessera d'accélérer son mouvement; quelle sera sa vites se complette; & combien elle parcourra de pieds avant que de l'acquérir. These tables, made by applying this hypothesis, show how many feet a lead ball six lines [1.3536 cm] in diameter will fall in each second; how many feet it will fall in any number of seconds we choose; when and where it stops accelerating; what will be its final speed; & how many feet it will cover before acquiring it. Adapted from Edme Mariotte, 1673

- Construct a graph of distance vs. time from Mariotte's predicted values and use it to determine the terminal velocity of his hypothetical lead ball.
- Use the contemporary drag equation, R = ½ρ CAv 2 , that evolved from Mariotte's hypothesis to determine the terminal velocity of his hypothetical lead ball.
- How do the results of your previous two analyses compare?

At terminal velocity, drag equals weight.

Replace those two symbols with appropriate contemporary equations.

Add subscripts to identify the two very different materials.

Relate the mass of the lead sphere to its density

1 2 ρ air C π r 2 v 2 = ρ lead 4 3 π r 3 g

1 2 ρ air Cv 2 = ρ lead 4 3 rg

List the quantities with numbers and units.

## practice problem 3

Acceleration is the rate of change of velocity with time.

Finish things off with a little bit of algebra…

This agrees with our condition that the skydiver wasn't moving at first.

One last test. What happens to our function if we let b = 0 ? What happens if we get rid of drag?

For our problem, where the limiting variable is b , we'll let…

So now we need to take the limit of this instead…

Repeat the previous approach using a drag that is proportional to speed squared.

Rearrange into a first order differential equation…

So what does that mean for us? Make the following replacement in the definition of tanh.

Look at the big beautiful pile of symbols we get.

## practice problem 4

Begin with the definition of work and play around with it a bit.

Replace the nonspecific force F with power law form of aerodynamic drag.

for drag calculations, not the computationally simple, but physically unrealistic…

but then, I don't know any automotive engineers.

- 5.2 Drag Forces
- Introduction to Science and the Realm of Physics, Physical Quantities, and Units
- 1.1 Physics: An Introduction
- 1.2 Physical Quantities and Units
- 1.3 Accuracy, Precision, and Significant Figures
- 1.4 Approximation
- Section Summary
- Conceptual Questions
- Problems & Exercises
- Introduction to One-Dimensional Kinematics
- 2.1 Displacement
- 2.2 Vectors, Scalars, and Coordinate Systems
- 2.3 Time, Velocity, and Speed
- 2.4 Acceleration
- 2.5 Motion Equations for Constant Acceleration in One Dimension
- 2.6 Problem-Solving Basics for One-Dimensional Kinematics
- 2.7 Falling Objects
- 2.8 Graphical Analysis of One-Dimensional Motion
- Introduction to Two-Dimensional Kinematics
- 3.1 Kinematics in Two Dimensions: An Introduction
- 3.2 Vector Addition and Subtraction: Graphical Methods
- 3.3 Vector Addition and Subtraction: Analytical Methods
- 3.4 Projectile Motion
- 3.5 Addition of Velocities
- Introduction to Dynamics: Newton’s Laws of Motion
- 4.1 Development of Force Concept
- 4.2 Newton’s First Law of Motion: Inertia
- 4.3 Newton’s Second Law of Motion: Concept of a System
- 4.4 Newton’s Third Law of Motion: Symmetry in Forces
- 4.5 Normal, Tension, and Other Examples of Forces
- 4.6 Problem-Solving Strategies
- 4.7 Further Applications of Newton’s Laws of Motion
- 4.8 Extended Topic: The Four Basic Forces—An Introduction
- Introduction: Further Applications of Newton’s Laws
- 5.1 Friction
- 5.3 Elasticity: Stress and Strain
- Introduction to Uniform Circular Motion and Gravitation
- 6.1 Rotation Angle and Angular Velocity
- 6.2 Centripetal Acceleration
- 6.3 Centripetal Force
- 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force
- 6.5 Newton’s Universal Law of Gravitation
- 6.6 Satellites and Kepler’s Laws: An Argument for Simplicity
- Introduction to Work, Energy, and Energy Resources
- 7.1 Work: The Scientific Definition
- 7.2 Kinetic Energy and the Work-Energy Theorem
- 7.3 Gravitational Potential Energy
- 7.4 Conservative Forces and Potential Energy
- 7.5 Nonconservative Forces
- 7.6 Conservation of Energy
- 7.8 Work, Energy, and Power in Humans
- 7.9 World Energy Use
- Introduction to Linear Momentum and Collisions
- 8.1 Linear Momentum and Force
- 8.2 Impulse
- 8.3 Conservation of Momentum
- 8.4 Elastic Collisions in One Dimension
- 8.5 Inelastic Collisions in One Dimension
- 8.6 Collisions of Point Masses in Two Dimensions
- 8.7 Introduction to Rocket Propulsion
- Introduction to Statics and Torque
- 9.1 The First Condition for Equilibrium
- 9.2 The Second Condition for Equilibrium
- 9.3 Stability
- 9.4 Applications of Statics, Including Problem-Solving Strategies
- 9.5 Simple Machines
- 9.6 Forces and Torques in Muscles and Joints
- Introduction to Rotational Motion and Angular Momentum
- 10.1 Angular Acceleration
- 10.2 Kinematics of Rotational Motion
- 10.3 Dynamics of Rotational Motion: Rotational Inertia
- 10.4 Rotational Kinetic Energy: Work and Energy Revisited
- 10.5 Angular Momentum and Its Conservation
- 10.6 Collisions of Extended Bodies in Two Dimensions
- 10.7 Gyroscopic Effects: Vector Aspects of Angular Momentum
- Introduction to Fluid Statics
- 11.1 What Is a Fluid?
- 11.2 Density
- 11.3 Pressure
- 11.4 Variation of Pressure with Depth in a Fluid
- 11.5 Pascal’s Principle
- 11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement
- 11.7 Archimedes’ Principle
- 11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
- 11.9 Pressures in the Body
- Introduction to Fluid Dynamics and Its Biological and Medical Applications
- 12.1 Flow Rate and Its Relation to Velocity
- 12.2 Bernoulli’s Equation
- 12.3 The Most General Applications of Bernoulli’s Equation
- 12.4 Viscosity and Laminar Flow; Poiseuille’s Law
- 12.5 The Onset of Turbulence
- 12.6 Motion of an Object in a Viscous Fluid
- 12.7 Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes
- Introduction to Temperature, Kinetic Theory, and the Gas Laws
- 13.1 Temperature
- 13.2 Thermal Expansion of Solids and Liquids
- 13.3 The Ideal Gas Law
- 13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
- 13.5 Phase Changes
- 13.6 Humidity, Evaporation, and Boiling
- Introduction to Heat and Heat Transfer Methods
- 14.2 Temperature Change and Heat Capacity
- 14.3 Phase Change and Latent Heat
- 14.4 Heat Transfer Methods
- 14.5 Conduction
- 14.6 Convection
- 14.7 Radiation
- Introduction to Thermodynamics
- 15.1 The First Law of Thermodynamics
- 15.2 The First Law of Thermodynamics and Some Simple Processes
- 15.3 Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
- 15.4 Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
- 15.5 Applications of Thermodynamics: Heat Pumps and Refrigerators
- 15.6 Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
- 15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation
- Introduction to Oscillatory Motion and Waves
- 16.1 Hooke’s Law: Stress and Strain Revisited
- 16.2 Period and Frequency in Oscillations
- 16.3 Simple Harmonic Motion: A Special Periodic Motion
- 16.4 The Simple Pendulum
- 16.5 Energy and the Simple Harmonic Oscillator
- 16.6 Uniform Circular Motion and Simple Harmonic Motion
- 16.7 Damped Harmonic Motion
- 16.8 Forced Oscillations and Resonance
- 16.10 Superposition and Interference
- 16.11 Energy in Waves: Intensity
- Introduction to the Physics of Hearing
- 17.2 Speed of Sound, Frequency, and Wavelength
- 17.3 Sound Intensity and Sound Level
- 17.4 Doppler Effect and Sonic Booms
- 17.5 Sound Interference and Resonance: Standing Waves in Air Columns
- 17.6 Hearing
- 17.7 Ultrasound
- Introduction to Electric Charge and Electric Field
- 18.1 Static Electricity and Charge: Conservation of Charge
- 18.2 Conductors and Insulators
- 18.3 Coulomb’s Law
- 18.4 Electric Field: Concept of a Field Revisited
- 18.5 Electric Field Lines: Multiple Charges
- 18.6 Electric Forces in Biology
- 18.7 Conductors and Electric Fields in Static Equilibrium
- 18.8 Applications of Electrostatics
- Introduction to Electric Potential and Electric Energy
- 19.1 Electric Potential Energy: Potential Difference
- 19.2 Electric Potential in a Uniform Electric Field
- 19.3 Electrical Potential Due to a Point Charge
- 19.4 Equipotential Lines
- 19.5 Capacitors and Dielectrics
- 19.6 Capacitors in Series and Parallel
- 19.7 Energy Stored in Capacitors
- Introduction to Electric Current, Resistance, and Ohm's Law
- 20.1 Current
- 20.2 Ohm’s Law: Resistance and Simple Circuits
- 20.3 Resistance and Resistivity
- 20.4 Electric Power and Energy
- 20.5 Alternating Current versus Direct Current
- 20.6 Electric Hazards and the Human Body
- 20.7 Nerve Conduction–Electrocardiograms
- Introduction to Circuits and DC Instruments
- 21.1 Resistors in Series and Parallel
- 21.2 Electromotive Force: Terminal Voltage
- 21.3 Kirchhoff’s Rules
- 21.4 DC Voltmeters and Ammeters
- 21.5 Null Measurements
- 21.6 DC Circuits Containing Resistors and Capacitors
- Introduction to Magnetism
- 22.1 Magnets
- 22.2 Ferromagnets and Electromagnets
- 22.3 Magnetic Fields and Magnetic Field Lines
- 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
- 22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications
- 22.6 The Hall Effect
- 22.7 Magnetic Force on a Current-Carrying Conductor
- 22.8 Torque on a Current Loop: Motors and Meters
- 22.9 Magnetic Fields Produced by Currents: Ampere’s Law
- 22.10 Magnetic Force between Two Parallel Conductors
- 22.11 More Applications of Magnetism
- Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
- 23.1 Induced Emf and Magnetic Flux
- 23.2 Faraday’s Law of Induction: Lenz’s Law
- 23.3 Motional Emf
- 23.4 Eddy Currents and Magnetic Damping
- 23.5 Electric Generators
- 23.6 Back Emf
- 23.7 Transformers
- 23.8 Electrical Safety: Systems and Devices
- 23.9 Inductance
- 23.10 RL Circuits
- 23.11 Reactance, Inductive and Capacitive
- 23.12 RLC Series AC Circuits
- Introduction to Electromagnetic Waves
- 24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed
- 24.2 Production of Electromagnetic Waves
- 24.3 The Electromagnetic Spectrum
- 24.4 Energy in Electromagnetic Waves
- Introduction to Geometric Optics
- 25.1 The Ray Aspect of Light
- 25.2 The Law of Reflection
- 25.3 The Law of Refraction
- 25.4 Total Internal Reflection
- 25.5 Dispersion: The Rainbow and Prisms
- 25.6 Image Formation by Lenses
- 25.7 Image Formation by Mirrors
- Introduction to Vision and Optical Instruments
- 26.1 Physics of the Eye
- 26.2 Vision Correction
- 26.3 Color and Color Vision
- 26.4 Microscopes
- 26.5 Telescopes
- 26.6 Aberrations
- Introduction to Wave Optics
- 27.1 The Wave Aspect of Light: Interference
- 27.2 Huygens's Principle: Diffraction
- 27.3 Young’s Double Slit Experiment
- 27.4 Multiple Slit Diffraction
- 27.5 Single Slit Diffraction
- 27.6 Limits of Resolution: The Rayleigh Criterion
- 27.7 Thin Film Interference
- 27.8 Polarization
- 27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light
- Introduction to Special Relativity
- 28.1 Einstein’s Postulates
- 28.2 Simultaneity And Time Dilation
- 28.3 Length Contraction
- 28.4 Relativistic Addition of Velocities
- 28.5 Relativistic Momentum
- 28.6 Relativistic Energy
- Introduction to Quantum Physics
- 29.1 Quantization of Energy
- 29.2 The Photoelectric Effect
- 29.3 Photon Energies and the Electromagnetic Spectrum
- 29.4 Photon Momentum
- 29.5 The Particle-Wave Duality
- 29.6 The Wave Nature of Matter
- 29.7 Probability: The Heisenberg Uncertainty Principle
- 29.8 The Particle-Wave Duality Reviewed
- Introduction to Atomic Physics
- 30.1 Discovery of the Atom
- 30.2 Discovery of the Parts of the Atom: Electrons and Nuclei
- 30.3 Bohr’s Theory of the Hydrogen Atom
- 30.4 X Rays: Atomic Origins and Applications
- 30.5 Applications of Atomic Excitations and De-Excitations
- 30.6 The Wave Nature of Matter Causes Quantization
- 30.7 Patterns in Spectra Reveal More Quantization
- 30.8 Quantum Numbers and Rules
- 30.9 The Pauli Exclusion Principle
- Introduction to Radioactivity and Nuclear Physics
- 31.1 Nuclear Radioactivity
- 31.2 Radiation Detection and Detectors
- 31.3 Substructure of the Nucleus
- 31.4 Nuclear Decay and Conservation Laws
- 31.5 Half-Life and Activity
- 31.6 Binding Energy
- 31.7 Tunneling
- Introduction to Applications of Nuclear Physics
- 32.1 Diagnostics and Medical Imaging
- 32.2 Biological Effects of Ionizing Radiation
- 32.3 Therapeutic Uses of Ionizing Radiation
- 32.4 Food Irradiation
- 32.5 Fusion
- 32.6 Fission
- 32.7 Nuclear Weapons
- Introduction to Particle Physics
- 33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited
- 33.2 The Four Basic Forces
- 33.3 Accelerators Create Matter from Energy
- 33.4 Particles, Patterns, and Conservation Laws
- 33.5 Quarks: Is That All There Is?
- 33.6 GUTs: The Unification of Forces
- Introduction to Frontiers of Physics
- 34.1 Cosmology and Particle Physics
- 34.2 General Relativity and Quantum Gravity
- 34.3 Superstrings
- 34.4 Dark Matter and Closure
- 34.5 Complexity and Chaos
- 34.6 High-temperature Superconductors
- 34.7 Some Questions We Know to Ask
- A | Atomic Masses
- B | Selected Radioactive Isotopes
- C | Useful Information
- D | Glossary of Key Symbols and Notation

## Learning Objectives

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

- Express mathematically the drag force.
- Discuss the applications of drag force.
- Define terminal velocity.
- Determine the terminal velocity given mass.

Using the equation for drag force, we have

Solving for the velocity, we obtain

## Take-Home Experiment

## Example 5.2

Find the terminal velocity of an 85-kg skydiver falling in a spread-eagle position.

Thus the terminal velocity v t v t can be written as

Using our equation for v t v t , we find that

## Stokes’ Law

## Galileo’s Experiment

As an Amazon Associate we earn from qualifying purchases.

Access for free at https://openstax.org/books/college-physics-2e/pages/1-introduction-to-science-and-the-realm-of-physics-physical-quantities-and-units

- Authors: Paul Peter Urone, Roger Hinrichs
- Publisher/website: OpenStax
- Book title: College Physics 2e
- Publication date: Jul 13, 2022
- Location: Houston, Texas
- Book URL: https://openstax.org/books/college-physics-2e/pages/1-introduction-to-science-and-the-realm-of-physics-physical-quantities-and-units
- Section URL: https://openstax.org/books/college-physics-2e/pages/5-2-drag-forces

## HIGH SCHOOL

## GRADUATE SCHOOL

- MCAT Tutoring
- GRE Tutoring
- LSAT Tutoring
- GMAT Tutoring
- AIMS Tutoring
- HSPT Tutoring
- ISEE Tutoring
- ISAT Tutoring
- SSAT Tutoring

## Search 50+ Tests

## math tutoring

## science tutoring

## elementary tutoring

## Search 350+ Subjects

- Video Overview
- Tutor Selection Process
- Online Tutoring
- Mobile Tutoring
- Instant Tutoring
- How We Operate
- Our Guarantee
- Impact of Tutoring
- Reviews & Testimonials
- Media Coverage
- About Varsity Tutors

## AP Physics C: Mechanics : Forces

## Example Question #22 : Forces

## Example Question #23 : Forces

A spherical asteroid has a hole drilled through the center as diagrammed below:

For the minimum energy case as the rocket leaves the surface:

Rearrange energy equation to isolate the velocity term.

Substitute in the given values to solve for the velocity.

## Example Question #2 : Calculating Gravitational Forces

Use the given values to solve for the force.

## Example Question #3 : Calculating Gravitational Forces

Newton's law of universal gravitation states:

We can write two equations for the gravity experienced before and after the doubling:

The equation for gravity after the doubling can be simplified:

## Example Question #4 : Calculating Gravitational Forces

Newton's law of universal gravitation states that:

Substituting these defintions into the second equation:

Substituting the definition of Fg1, we see:

Thus the gravitational forces doubles when the mass of one object doubles.

## Example Question #5 : Calculating Gravitational Forces

What does the spring scale read when the elevator is descending at constant speed?

## Example Question #6 : Calculating Gravitational Forces

(Note: You can treat the moon as a smooth sphere, and assume there’s no atmosphere.)

If we plug everything in, we get

## Example Question #24 : Forces

We can use Newton's second law:

Set up equations for the force on the moon and the force on Earth:

## Report an issue with this question

## DMCA Complaint

Please follow these steps to file a notice:

You must include the following:

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC 101 S. Hanley Rd, Suite 300 St. Louis, MO 63105

## Contact Information

## Find the Best Tutors

## 6 Applications of Newton’s Laws

6.4 drag force and terminal speed, 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

## Drag Forces

Figure 6.30 NASA researchers test a model plane in a wind tunnel. (credit: NASA/Ames)

## Terminal Velocity

Using the equation for drag force, we have

Solving for the velocity, we obtain

## Terminal Velocity of a Skydiver

Find the terminal velocity of an 85-kg skydiver falling in a spread-eagle position.

The terminal velocity [latex] {v}_{\text{T}} [/latex] can be written as

## Significance

## Check Your Understanding

Find the terminal velocity of a 50-kg skydiver falling in spread-eagle fashion.

## Stokes’ Law

For a spherical object falling in a medium, the drag force is

## The Calculus of Velocity-Dependent Frictional Forces

Figure 6.33 Free-body diagram of an object falling through a resistive medium.

Assuming that [latex] v=0\,\text{at}\,t=0, [/latex] integration of this equation yields

Assuming [latex] y=0\,\text{when}\,t=0, [/latex]

## Effect of the Resistive Force on a Motorboat

which, since [latex] \text{ln}A=x\,\text{implies}\,{e}^{x}=A, [/latex] we can write this as

Now from the definition of velocity,

With the initial position zero, we have

Now the boat’s limiting position is

- Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity in air, the drag force is determined using the drag coefficient (typical values are given in (Figure) ), the area of the object facing the fluid, and the fluid density.
- For small objects (such as a bacterium) moving in a denser medium (such as water), the drag force is given by Stokes’ law.

## Key Equations

By what factor does the drag force on a car increase as it goes from 65 to 110 km/h?

[latex] {(\frac{110}{65})}^{2}=2.86 [/latex] times

Using Stokes’ law, verify that the units for viscosity are kilograms per meter per second.

[latex] 0.76\,\text{kg/m}·\text{s} [/latex]

As shown below, if [latex] M=5.50\,\text{kg,} [/latex] what is the tension in string 1?

## Additional Problems

a. 0.186 N; b. 774 N; c. 0.48 N

## Challenge Problems

[latex] v=\sqrt{{v}_{0}{}^{2}-2g{r}_{0}(1-\frac{{r}_{0}}{r})} [/latex]

a. 53.9 m/s; b. 328 m; c. 4.58 m/s; d. 257 s

a. [latex] v=20.0(1-{e}^{-0.01t}); [/latex] b. [latex] {v}_{\text{limiting}}=20\,\text{m/s} [/latex]

## Real World Applications — for high school level and above

## Education & Theory — for high school level and above

## Kids Section

- Physics For Kids
- Science Experiments
- Science Fair Ideas
- Science Quiz
- Science Toys
- Teacher Resources
- Commercial Disclosure
- Privacy Policy

© Copyright 2009-2023 real-world-physics-problems.com

- school Campus Bookshelves
- menu_book Bookshelves
- perm_media Learning Objects
- login Login
- how_to_reg Request Instructor Account
- hub Instructor Commons
- Download Page (PDF)
- Download Full Book (PDF)
- Periodic Table
- Physics Constants
- Scientific Calculator
- Reference & Cite
- Tools expand_more
- Readability

selected template will load here

## 6.7: Drag Force and Terminal Speed

## Learning Objectives

- Express the drag force mathematically
- Describe applications of the drag force
- Define terminal velocity
- Determine an object’s terminal velocity given its mass

## Drag Forces

\[F_{D} = \frac{1}{2} C \rho A v^{2}, \label{6.5}\]

## Definition: Drag Force

Drag force \(F_D\) is proportional to the square of the speed of the object. Mathematically,

\[F_{D} = \frac{1}{2} C \rho A v^{2},\]

## Terminal Velocity

\[F_{net} = mg - F_{D} = ma = 0 \ldotp\]

Using the equation for drag force, we have

\[mg = \frac{1}{2} C \rho A v_{T}^{2} \ldotp\]

Solving for the velocity, we obtain

\[v_{T} = \sqrt{\frac{2mg}{\rho CA}} \ldotp\]

## Example \(\PageIndex{1}\): Terminal Velocity of a Skydiver

Find the terminal velocity of an 85-kg skydiver falling in a spread-eagle position.

The terminal velocity \(v_T\) can be written as

## Exercise \(\PageIndex{1}\)

Find the terminal velocity of a 50-kg skydiver falling in spread-eagle fashion.

“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.”

## Stokes’ law

For a spherical object falling in a medium, the drag force is

\[F_{s} = 6 \pi r \eta v, \label{6.6}\]

Video \(\PageIndex{1}\) : Fluid Mechanics - Drag force - Flow simulation

## The Calculus of Velocity-Dependent Frictional Forces

\[mg - bv = m \frac{dv}{dt},\]

\[v_{T} = \frac{mg}{b} \ldotp\]

\[\frac{dv}{g- \left(\dfrac{b}{m}\right)v} = dt \ldotp \label{eq20}\]

Assuming that \(v = 0\) at \9t = 0\), integration of Equation \ref{eq20} yields

\[\int_{0}^{v} \frac{dv'}{g- \left(\dfrac{b}{m}\right)v'} = \int_{0}^{t} dt',\]

\[- \frac{m}{b} \ln \left(g - \dfrac{b}{m} v' \right) \Bigg|_{0}^{v} = t' \big|_{0}^{t} ,\]

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

\[- \frac{m}{b} [ \ln \left(g - \dfrac{b}{m} v \right) - \ln g] = t \ldotp\]

\[\frac{g - \left(\dfrac{bv}{m}\right)}{g} = e^{- \frac{bt}{m}},\]

\[v = \frac{mg}{b} \big( 1 - e^{- \frac{bt}{m}} \big) \ldotp\]

Notice that as t → \(\infty\), v → \(\frac{mg}{b}\) = v T , which is the terminal velocity.

The position at any time may be found by integrating the equation for v. With v = \(\frac{dy}{dt}\),

\[dy = \frac{mg}{b} \big( 1 - e^{- \frac{bt}{m}} \big)dt \ldotp\]

\[\int_{0}^{y} dy' = \frac{mg}{b} \int_{0}^{t} \big( 1 - e^{- \frac{bt}{m}} \big)dt',\]

\[y = \frac{mg}{b} t + \frac{m^{2}g}{b^{2}} \big( e^{- \frac{bt}{m}} - 1 \big) \ldotp\]

## Example \(\PageIndex{2}\): Effect of the Resistive Force on a Motorboat

- What are the velocity and position of the boat as functions of time?
- If the boat slows down from 4.0 to 1.0 m/s in 10 s, how far does it travel before stopping?
- With the motor stopped, the only horizontal force on the boat is f R = −bv, so from Newton’s second law, $$m \frac{dv}{dt} = -bv,$$which we can write as $$\frac{dv}{v} = - \frac{b}{m} dt \ldotp$$Integrating this equation between the time zero when the velocity is v 0 and the time t when the velocity is v, we have $$\int_{0}^{v} \frac{dv'}{v'} = -\frac{b}{m} \int_{0}^{t} dt' \ldotp$$ Thus, $$\ln \frac{v}{v_{0}} = - \frac{b}{m} t,$$which, since lnA = x implies e x = A, we can write this as $$v = v_{0} e^{- \frac{bt}{m}} \ldotp$$Now from the definition of velocity, $$\frac{dx}{dt} = v_{0} e^{- \frac{bt}{m}},$$so we have $$dx = v_{0} e^{- \frac{bt}{m}} dt \ldotp$$With the initial position zero, we have $$\int_{0}^{x} dx' = v_{0} \int_{0}^{t} e^{- \frac{bt'}{m}} dt' ,$$and $$x = - \frac{mv_{0}}{b} e^{- \frac{bt}{m}} \Big|_{0}^{t} = \frac{mv_{0}}{b} \big(1 - e^{- \frac{bt}{m}} \big) \ldotp$$As time increases, \(e^{- \frac{bt}{m}}\) → 0, and the position of the boat approaches a limiting value $$x_{max} = \frac{mv_{0}}{b} \ldotp$$Although this tells us that the boat takes an infinite amount of time to reach x max , the boat effectively stops after a reasonable time. For example, at t = \(\frac{10m}{b}\), we have$$v = v_{0} e^{-10} \simeq 4.5 \times 10^{-5} v_{0},$$whereas we also have $$x = x_{max} \big(1 - e^{-10} \big) \simeq 0.99995x_{max} \ldotp$$Therefore, the boat’s velocity and position have essentially reached their final values.
- With v 0 = 4.0 m/s and v = 1.0 m/s, we have 1.0 m/s = (4.0 m/s) \(e^{(- \frac{bt}{m})(10\; s)}\), so $$\ln 0.25 = - \ln 4.0 = - \frac{b}{m} (10\;s),$$and $$\frac{b}{m} = \frac{1}{10} \ln 4.0\; s^{-1} = 0.14\; s^{-1} \ldotp$$Now the boat's limiting position is $$x_{max} = \frac{mv_{0}}{b} = \frac{4.0\; m/s}{0.14\; s^{-1}} = 29\; m \ldotp$$

## Exercise \(\PageIndex{2}\)

## Drag Force Formula

## Drag Coefficient Formula

Following is the formula used to calculate the drag coefficient:

- C d is the drag coefficient
- ρ is the density of the medium in kg.m -3
- V is the velocity of the body in km.h -1
- A is the cross-sectional area in m 2

## Solved Examples

Solution: Given: Velocity, V= 80 km.h -1

Cross-sectional area, A= 6 m 2

Density of fluid, ρ =1.2 kg.m -3

Density of fluid, ρ=1.2 kg.m -3

Cross-sectional area, A=110 m 2

Stay tuned with BYJU’S to learn more about other Physics related concepts.

## Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

## Register with BYJU'S & Download Free PDFs

## Select your language

- Absolute Magnitude
- Astronomical Objects
- Astronomical Telescopes
- Black Body Radiation
- Classification by Luminosity
- Classification of Stars
- Doppler Effect
- Exoplanet Detection
- Hertzsprung-Russell Diagrams
- Hubble's Law
- Large Diameter Telescopes
- Radio Telescopes
- Reflecting Telescopes
- Stellar Spectral Classes
- Fission and Fusion
- Medical Tracers
- Nuclear Reactors
- Radiotherapy
- Random Nature of Radioactive Decay
- Thickness Monitoring
- Applications of Circular Motion
- Centripetal and Centrifugal Force
- Circular Motion and Free-Body Diagrams
- Fundamental Forces
- Gravitational and Electric Forces
- Gravity on Different Planets
- Inertial and Gravitational Mass
- Vector Fields
- Spring Mass System
- Application of Newton's Second Law
- Dynamic Systems
- Free Body Diagrams
- Friction Force
- Normal Force
- Springs Physics
- Superposition of Forces
- Charge Distribution
- Charged Particle in Uniform Electric Field
- Conservation of Charge
- Electric Field Between Two Parallel Plates
- Electric Field Lines
- Electric Field of Multiple Point Charges
- Electric Force
- Electric Potential Due to Dipole
- Electric Potential due to a Point Charge
- Electrical Systems
- Equipotential Lines
- Attraction and Repulsion
- Basics of Electricity
- Capacitors in Series and Parallel
- Circuit Schematic
- Circuit Symbols
- Current Density
- Current-Voltage Characteristics
- Electric Current
- Electric Motor
- Electrical Power
- Electricity Generation
- Emf and Internal Resistance
- Kirchhoff's Junction Rule
- Kirchhoff's Loop Rule
- National Grid Physics
- Potential Difference
- Power Rating
- Resistance and Resistivity
- Resistivity
- Resistors in Series and Parallel
- Series and Parallel Circuits
- Simple Circuit
- Static Electricity
- Superconductivity
- Time Constant of RC Circuit
- Transformer
- Voltage Divider
- Benjamin Franklin's Kite Experiment
- Changing Magnetic Field
- Circuit Analysis
- Diamagnetic Levitation
- Electric Dipole
- Electric Field Energy
- Oersted's Experiment
- Ampere's Law
- Lorentz Force Law
- Magnetic Moment
- Gauss's Law
- Permittivity
- Big Energy Issues
- Conservative and Non Conservative Forces
- Efficiency in Physics
- Elastic Potential Energy
- Electrical Energy
- Energy and the Environment
- Forms of Energy
- Geothermal Energy
- Gravitational Potential Energy
- Heat Engines
- Heat Transfer Efficiency
- Kinetic Energy
- Mechanical Power
- Potential Energy
- Potential Energy and Energy Conservation
- Pulling Force
- Renewable Energy Sources
- Wind Energy
- Work Energy Principle
- Angular Momentum
- Angular Work and Power
- Engine Cycles
- First Law of Thermodynamics
- Moment of Inertia
- Non-Flow Processes
- PV Diagrams
- Reversed Heat Engines
- Rotational Kinetic Energy
- Second Law and Engines
- Thermodynamics and Engines
- Torque and Angular Acceleration
- Alternating Currents
- Capacitance
- Capacitor Charge
- Capacitor Discharge
- Coulomb's Law
- Electric Field Strength
- Electric Fields
- Electric Potential
- Electromagnetic Induction
- Energy Stored by a Capacitor
- Escape Velocity
- Gravitational Field Strength
- Gravitational Fields
- Gravitational Potential
- Magnetic Fields
- Magnetic Flux Density
- Magnetic Flux and Magnetic Flux Linkage
- Moving Charges in a Magnetic Field
- Newton’s Laws
- Operation of a Transformer
- Parallel Plate Capacitor
- Planetary Orbits
- Synchronous Orbits
- Absolute Pressure and Gauge Pressure
- Application of Bernoulli's Equation
- Archimedes' Principle
- Conservation of Energy in Fluids
- Fluid Systems
- Force and Pressure
- Air resistance and friction
- Conservation of Momentum
- Contact Forces
- Elastic Forces
- Force and Motion
- Impact Forces
- Moment Physics
- Moments Levers and Gears
- Moments and Equilibrium
- Resultant Force
- Safety First
- Time Speed and Distance
- Velocity and Acceleration
- Error Calculation
- Unit Conversion
- Writing a Lab Report
- Bottle Rocket
- Charles law
- Circular Motion
- Diesel Cycle
- Heat Transfer
- Heat Transfer Experiments
- Ideal Gas Model
- Ideal Gases
- Kinetic Theory of Gases
- Models of Gas Behaviour
- Newton's Law of Cooling
- Periodic Motion
- Rankine Cycle
- Simple Harmonic Motion
- Simple Harmonic Motion Energy
- Temperature
- Thermal Equilibrium
- Thermal Physics
- Work in Thermodynamics
- Dispersion of Light
- Periodic Wave
- Reflection at Spherical Surfaces
- Refractive Index
- Air Resistance
- Angular Kinematic Equations
- Average Velocity and Acceleration
- Displacement, Time and Average Velocity
- Frame of Reference
- Free Falling Object
- Kinematic Equations
- Motion in One Dimension
- Motion in Two Dimensions
- Rotational Motion
- Uniformly Accelerated Motion
- Center of Gravity
- Change of Momentum
- Momentum Change and Impulse
- Ampere force
- Earth's Magnetic Field
- Fleming's Left Hand Rule
- Induced Potential
- Magnetic Forces and Fields
- Motor Effect
- Particles in Magnetic Fields
- Permanent and Induced Magnetism
- Faraday's Law
- Induced Currents
- Magnetic Field of a Current-Carrying Wire
- Magnetic Flux
- Magnetic Materials
- Monopole vs Dipole
- Estimation of Errors
- Limitations of Measurements
- Uncertainty and Errors
- Acceleration Due to Gravity
- Bouncing Ball Example
- Bulk Properties of Solids
- Centre of Mass
- Collisions and Momentum Conservation
- Conservation of Energy
- Elastic Collisions
- Force Energy
- Graphs of Motion
- Linear Motion
- Materials Energy
- Power and Efficiency
- Projectile Motion
- Scalar and Vector
- Terminal Velocity
- Vector Problems
- Work and Energy
- Young's Modulus
- Absorption of X-Rays
- CT Scanners
- Defects of Vision
- Defects of Vision and Their Correction
- Diagnostic X-Rays
- Effective Half Life
- Electrocardiography
- Fibre Optics and Endoscopy
- Gamma Camera
- Hearing Defects
- High Energy X-Rays
- Magnetic Resonance Imaging
- Noise Sensitivity
- Non Ionising Imaging
- Physics of Vision
- Physics of the Ear
- Physics of the Eye
- Radioactive Implants
- Radionuclide Imaging Techniques
- Radionuclide Imaging and Therapy
- Structure of the Ear
- Ultrasound Imaging
- X-Ray Image Processing
- X-Ray Imaging
- Bohr Model of the Atom
- Disintegration Energy
- Franck Hertz Experiment
- Mass Energy Equivalence
- Nucleus Structure
- Quantization of Energy
- Spectral Lines
- The Discovery of the Atom
- Wave Function
- Alpha Beta and Gamma Radiation
- Binding Energy
- Induced Fission
- Mass and Energy
- Nuclear Instability
- Nuclear Radius
- Radioactive Decay
- Radioactivity
- Rutherford Scattering
- Safety of Nuclear Reactors
- Energy Time Graph
- Energy in Simple Harmonic Motion
- Kinetic Energy in Simple Harmonic Motion
- Mechanical Energy in Simple Harmonic Motion
- Period of Pendulum
- Period, Frequency and Amplitude
- Phase Angle
- Physical Pendulum
- Restoring Force
- Simple Pendulum
- Spring-Block Oscillator
- Torsional Pendulum
- Changes of state
- Gas Pressure and Temperature
- Internal Energy
- Specific Latent Heat
- Volume of Gas
- Converting Units
- Physical Quantities
- SI Prefixes
- Standard Form Physics
- Units Physics
- Use of SI Units
- Acceleration
- Angular Acceleration
- Angular Displacement
- Angular Velocity
- Centrifugal Force
- Centripetal Force
- Displacement
- Equilibrium
- Forces of Nature Physics
- Galileo's Leaning Tower of Pisa Experiment
- Inclined Plane
- Mass in Physics
- Speed Physics
- Static Equilibrium
- Antiparticles
- Atomic Model
- Classification of Particles
- Collisions of Electrons with Atoms
- Conservation Laws
- Electromagnetic Radiation and Quantum Phenomena
- Neutron Number
- Quark Physics
- Specific Charge
- The Photoelectric Effect
- Wave-Particle Duality
- Angular Impulse
- Angular Kinematics
- Angular Motion and Linear Motion
- Connecting Linear and Rotational Motion
- Orbital Trajectory
- Rotational Equilibrium
- Rotational Inertia
- Satellite Orbits
- Third Law of Kepler
- Data Collection
- Data Representation
- Drawing Conclusions
- Equations in Physics
- Uncertainties and Evaluations
- Orbital Motions
- The Life Cycle of a Star
- Heat Radiation
- Thermal Conductivity
- Thermal Efficiency
- Thermodynamic Diagram
- Thermodynamic Force
- Thermodynamic and Kinetic Control
- Centripetal Acceleration and Centripetal Force
- Conservation of Angular Momentum
- Force and Torque
- Muscle Torque
- Newton's Second Law in Angular Form
- Simple Machines
- Unbalanced Torque
- Centripetal Force and Velocity
- Critical Speed
- Free Fall and Terminal Velocity
- Gravitational Acceleration
- Gravitational Force
- Kinetic Friction
- Object in Equilibrium
- Orbital Period
- Resistive Force
- Spring Force
- Static Friction
- Cathode Rays
- Discovery of the Electron
- Einstein's Theory of Special Relativity
- Electromagnetic Waves
- Electron Microscopes
- Electron Specific Charge
- Length Contraction
- Michelson-Morley Experiment
- Millikan's Experiment
- Newton's and Huygens' Theories of Light
- Photoelectricity
- Relativistic Mass and Energy
- Special Relativity
- Thermionic Electron Emission
- Time Dilation
- Wave Particle Duality of Light
- Applications of Ultrasound
- Applications of Waves
- Diffraction
- Diffraction Gratings
- Doppler Effect in Light
- Earthquake Shock Waves
- Echolocation
- Image Formation by Lenses
- Interference
- Longitudinal Wave
- Longitudinal and Transverse Waves
- Oscilloscope
- Phase Difference
- Polarisation
- Progressive Waves
- Properties of Waves
- Ray Diagrams
- Ray Tracing Mirrors
- Refraction at a Plane Surface
- Resonance in Sound Waves
- Seismic Waves
- Snell's law
- Standing Waves
- Stationary Waves
- Total Internal Reflection in Optical Fibre
- Transverse Wave
- Wave Characteristics
- Waves in Communication
- Conservative Forces and Potential Energy
- Dissipative Force
- Energy Dissipation
- Energy in Pendulum
- Force and Potential Energy
- Force vs. Position Graph
- Orbiting Objects
- Potential Energy Graphs and Motion
- Spring Potential Energy
- Total Mechanical Energy
- Translational Kinetic Energy
- Work Energy Theorem
- Work and Kinetic Energy

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Nie wieder prokastinieren mit unseren Lernerinnerungen.

## Drag Force in Physics

In physics, drag force is the force that opposes the relative motion between an object and a fluid.

## Types of Drag Force

There are different types of drag force, especially when considering the flight of airplanes:

- Form Drag — The drag due to the shape of the object moving through the fluid.
- Skin Friction Drag — The drag due to the roughness of the object's surface.
- Interference Drag — The drag resulting from two airflows of different speeds meeting and interfering.
- Induced Drag — The drag resulting from lift.
- Wave Drag — The drag due to shockwaves.

## Examples of Drag Force

Here we go again — even more examples. It's like walking through honey.

- Any object falling through the air. For example, skydivers use principles of drag force to move and position themselves through the air, and when they open their parachutes, the greater drag force that's created helps slow them down to land.
- Drag force slows down cars, planes, and ships when they move. So engineers increase the aerodynamics of these vehicles to reduce drag and increase the vehicles' efficiency.
- Swimmers fight against drag force when they swim. Even shaving can reduce drag and increase swimmers' speeds.
- Flying squirrels use their wing-like skin to use drag force to control their flight and landings.
- If you were to stand on top of a quickly moving train, it wouldn't be as easy to stand or run as many movies make it seem because there would be a drag force pushing against you.
- Drag force causes kites to fly. You must run forward with the kite to get the drag force to push against it and lift it into the air.

## Drag Force Equation

The common equation or formula for drag force is shown below:

$$D=\frac{1}{2}\\C\rho Av^2\mathrm{.}$$

## Drag Force Formula

## Stokes's Law

## Drag Force in Free-Fall — Terminal Velocity

## Example Problem Using Drag Force

Now, it's time for an example.

$$D=\frac{1}{2}\\ C\rho A v^2\mathrm{,}$$

$$D=\frac{1}{2}\\(2.1)(1.225\,\mathrm{kg/m^3})(0.25\,\mathrm{m^2})(12\,\mathrm{m/s^2})^2\mathrm{,}$$

## Drag Force - Key takeaways

- The drag force is the force that opposes the relative motion between an object and a fluid.
- The direction of the drag force is always opposite to the relative motion.
- Common types of drag force include parasitic drag, form drag, skin friction drag, interference drag, induced drag, and wave drag.
- For most simple scenarios (if the velocity is high, the viscosity of the fluid is low, and the object isn't tiny), the equation for drag force is \(D=\frac{1}{2}\\C\rho Av^2\).
- We can use Stokes's Law to find the drag force when a situation doesn't meet the requirements necessary to use the drag force.

## Frequently Asked Questions about Drag Force

Drag force is the force that opposes the relative motion between an object and a fluid.

## --> How to calculate drag force?

## --> What are the types of drag force?

## --> What is the importance of drag force?

## --> What is viscous drag force?

## Final Drag Force Quiz

The force that opposes the motion between an object and a fluid.

In what direction does the drag force act?

Opposite to the relative motion.

What condition does not apply to being able to use the drag force equation accurately?

The coefficient of drag is less than \(2\).

Which of the following is not an example of drag force?

A box resists being pulled across the floor with a rope.

What does \(A\) represent in the drag force equation?

The effective cross-sectional area of the object.

If the density of the fluid increases, the drag force decreases.

If the relative velocity between the object and fluid increases, the drag force increases.

If the effective cross-sectional area of the object decreases, the drag force increases.

Objects with a higher drag coefficient will create more drag than those with a lower coefficient.

Bob wants to reduce drag to get better miles per gallon on his car. What might he do?

of the users don't pass the Drag Force quiz! Will you pass the quiz?

## More explanations about Dynamics

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

## Study Analytics

Identify your study strength and weaknesses.

## Weekly Goals

Set individual study goals and earn points reaching them.

## Smart Reminders

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

## Magic Marker

Create flashcards in notes completely automatically.

## Smart Formatting

Create the most beautiful study materials using our templates.

## Join millions of people in learning anywhere, anytime - every day

Sign up to highlight and take notes. It’s 100% free.

## This is still free to read, it's not a paywall.

You need to register to keep reading, get free access to all of our study material, tailor-made.

Over 10 million students from across the world are already learning smarter.

## IMAGES

## VIDEO

## COMMENTS

FOS4 – Practice Problems –Drag Force – APC. 1) A 0.500-kg object is suspended from the ceiling of an accelerating boxcar as shown.

Practice. practice problem 1. Two related questions… Determine the drag coefficient of a 75 kg skydiver

Balanced Forces Equilibrium Example · Drag Force Differential Equation · Newton's Law of Motion - First, Second & Third - Physics · Centripetal

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).

assesses the students' ability to apply calculus in the solution of a physics problem.

An object of mass \displaystyle \small m is dropped from a tower. The object's drag force is given by \displaystyle \small F_{d

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

Drag Force. A drag force is the resistance force caused by the motion of a body through a fluid, such as water or air. A drag force acts opposite to the

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

Question 1. A car travels with a speed of 80 km.h-1 with a drag coefficient of 0.25. If the cross-sectional area is 6

Example Problem Using Drag Force.