<< Chapter < Page Chapter >> Page >
A photograph of geese flying in a V formation.
Geese fly in a V formation during their long migratory travels. This shape reduces drag and energy consumption for individual birds, and also allows them a better way to communicate. (credit: “Julo”/Wikimedia Commons)

In lecture demonstrations, we do measurements of the drag force on different objects. The objects are placed in a uniform airstream created by a fan. Calculate the Reynolds number and the drag coefficient.

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 . 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

f R = b v ,

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 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 [link] . Newton’s second law in the vertical direction gives the differential equation

m g b v = m d v d t ,

where we have written the acceleration as 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 . From the previous equation,

m g b v T = 0 ,


v T = m g b .
The free body diagram shows forces m times vector g pointing vertically down and b times vector v pointing vertically up. The velocity, vector v, is vertically down. The positive y direction is also vertically down.
Free-body diagram of an object falling through a resistive medium.

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 .

Assuming that v = 0 at t = 0 , integration of this equation yields

0 v d v g ( b / m ) v = 0 t d t ,


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

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

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

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

g ( b v / m ) g = e b t / m ,


v = m g b ( 1 e b t / m ) .

Notice that as 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 ,

d y = m g b ( 1 e b t / m ) d t .

Assuming y = 0 when t = 0 ,

0 y d y = m g b 0 t ( 1 e b t ' / m ) d t ,

which integrates to

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

Effect of the resistive force on a motorboat

A motorboat is moving across a lake at a speed v 0 when its motor suddenly freezes up and stops. The boat then slows down under the frictional force f R = b v . (a) What are the velocity and position of the boat as functions of time? (b) If the boat slows down from 4.0 to 1.0 m/s in 10 s, how far does it travel before stopping?


  1. With the motor stopped, the only horizontal force on the boat is f R = b v , so from Newton’s second law,
    m d v d t = b v ,

    which we can write as
    d v v = b m d t .

    Integrating this equation between the time zero when the velocity is v 0 and the time t when the velocity is v , we have
    0 v d v v = b m 0 t d t .

    ln v v 0 = b m t ,

    which, since ln A = x implies e x = A , we can write this as
    v = v 0 e b t / m .

    Now from the definition of velocity,
    d x d t = v 0 e b t / m ,

    so we have
    d x = v 0 e b t / m d t .

    With the initial position zero, we have
    0 x d x ' = v 0 0 t e b t ' / m d t ' ,

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

    As time increases, e b t / m 0 , and the position of the boat approaches a limiting value
    x max = m v 0 b .

    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 = 10 m / b , we have
    v = v 0 e −10 4.5 × 10 −5 v 0 ,

    whereas we also have
    x = x max ( 1 e −10 ) 0.99995 x max .

    Therefore, the boat’s velocity and position have essentially reached their final values.
  2. With v 0 = 4.0 m/s and v = 1.0 m/s, we have 1.0 m/s = ( 4.0 m/s ) e ( b / m ) ( 10 s ) , so
    ln 0.25 = ln 4.0 = b m ( 10 s ) ,

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

    Now the boat’s limiting position is
    x max = m v 0 b = 4.0 m/s 0.14 s −1 = 29 m .

Questions & Answers

explain equilibrium of a body
pls who can tell me more about Kirchoff's law?
Hi all, love you all!!!
Giorgi Reply
How does resonance occur
Rahim Reply
plzz explanation of the equilibrium of a body
what is quantam
pamit Reply
quantum is a division of mechanics
what is friction
Muhammad Reply
a force act by surface between two bodies whose are always oppose the relative motion .....
when two rough bodies are placed in contact and try to slip each other ... than a force act them and it's ippse the relative motion between them
thats friction force and roughnes of both bodies is define friction of surface
what is a progressive wave
sheriff-deen Reply
What is the wake for therapist
Ife Reply
can u like explain your question with clear detail
who would teach me vectors?
Tintin Reply
are you guys are physics student?
what u want to know in vectors
I vl help, I m tutor
yes sir
calculations get me confused sometimes
what's chemistry
Esther Reply
branch of science dt deals with the study of physical properties of matter and it's particulate nature
Y acctually do u hav ur way of defining it? just bring ur iwn idear
well, it deals with the weight of substances and reaction behind them as well as the behavior
buh hope Esther, we've answered ur question
what's ohms law
ohms law states that, the current flowing through an electric circuit is directly proportional to the potential difference, provided temperature and pressure are kept constant
what is sound
ohms law states that the resistance of a material is directly proportional to the potential difference between two points on that material, if temperature and other physical conditions become constant
How do I access the MCQ
Abraham Reply
As I think the best is, first select the easiest questions for you .and then you can answer the remaining questions.
I mean I'm unable to view it
when I click on it, it doesn't respond
ohhh,try again and again ,It will be showed
what is centripetal force
Don Reply
هي قوة ناتجة من الحركة الدائرية ويكون اتجاهها إلى المركز دائماً
meaning of vector quantity
Felix Reply
vector quantity is any quantity that has both magnitude in terms of number (units) and direction in terms of viewing the quantity from an origin using angles (degree) or (NEWS) method
vector quantity is physical quantity has magnitude and direction
vector is a quantity that is use in measuring size of physical properties and their direction
what difference and similarities between work,force,energy and power?
Anes Reply
I need the best answer
enery is the ability to do work. work is job done, force is a pull or push. power has to do with potential. they belong to different categories which include heat energy, electricity.
force refers to a push or pull... energy refers to work done while power is work done per unit time
mathematically express angular velocity and angular acceleration
Mario Reply
it depends on the direction. an angular velocity will be linear and angular acceleration will be an angle of elevation.
The sonic range finder discussed in the preceding question often needs to be calibrated. During the calibration, the software asks for the room temperature. Why do you suppose the room temperature is required?
Shaina Reply
Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See [link] .) (a) Calculate the echo times for temperatures of 5.00°C5.00°C and 35.0°C.35.0°C. (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty an
give a reason why musicians commonly bring their wind instruments to room temperature before playing them.
The ear canal resonates like a tube closed at one end. (See [link]Figure 17_03_HumEar[/link].) If ear canals range in length from 1.80 to 2.60 cm in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be 37.0°C,37.0°C, which is the same as body tempe
By what fraction will the frequencies produced by a wind instrument change when air temperature goes from 10.0°C10.0°C to 30.0°C30.0°C ? That is, find the ratio of the frequencies at those temperatures.
Practice Key Terms 2

Get the best University physics vol... course in your pocket!

Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?