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The motion of an aircraft through the air can be explained and described by physical

principals discovered over 300 years ago by Sir Isaac Newton. Newton worked in many

areas of mathematics and physics. e developed the theories of gravitation in !"""#

when he was only $3 years old. Some twenty years later# in !"%"# he presented his

three laws of motion in the &'rincipia (athematica 'hilosophiae Naturalis.& The laws are

shown above# and the application of these laws to aerodynamics are given on separate

slides.

Newton)s first law states that every ob*ect will remain at rest or in uniform motion in a

straight line unless compelled to change its state by the action of an external force. This

is normally taken as the definition of inertia. The key point here is that if there is no netforce acting on an ob*ect +if all the external forces cancel each other out, then the

ob*ect will maintain a constant velocity. If that velocity is -ero# then the ob*ect remains

at rest. If an external force is applied# the velocity will change because of the force.

The second law explains how the velocity of an ob*ect changes when it is sub*ected to

an external force. The law defines aforce to be eual to change in momentum +mass

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times velocity, per change in time. Newton also developed the calculus of mathematics#

and the &changes& expressed in the second law are most accurately defined in

differential forms. +/alculus can also be used to determine the velocity and location

variations experienced by an ob*ect sub*ected to an external force., or an ob*ect with a

constant mass m# the second law states that the force F is the product of an ob*ect)s

mass and its acceleration a1

2 m a

or an external applied force# the change in velocity depends on the mass of the ob*ect.

 4 force will cause a change in velocity5 and likewise# a change in velocity will generate a

force. The euation works both ways.

The third law states that for every action +force, in nature there is an eual and opposite

reaction. In other words# if ob*ect 4 exerts a force on ob*ect 6# then ob*ect 6 also exerts

an eual force on ob*ect 4. Notice that the forces are exerted on different ob*ects. The

third law can be used to explain the generation of lift by a wing and the production

of thrust by a *et engine.

You can view a short movie of "Orville and Wilbur Wright" explaining how Newton's

Laws of Motion described the flight of their aircraft. he movie file can be saved to !our

computer and viewed as a odcast on !our podcast pla!er.

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Sir Isaac Newton first presented his three laws of motion in the &'rincipia (athematica

'hilosophiae Naturalis& in !"%". is first law states that every ob*ect will remain at rest

or in uniform motion in a straight line unless compelled to change its state by the action

of an external force. This is normally taken as the definition of inertia. The key pointhere is that if there is no net force resulting from unbalanced forces acting on an ob*ect

+if all the external forces cancel each other out,# then the ob*ect will maintain a constant

velocity. If that velocity is -ero# then the ob*ect remains at rest. 4nd if an additional

external force is applied# the velocity will change because of the force. The amount of

the change in velocity is determined by Newton)s second law of motion.

There are many excellent examples of Newton)s first law involving aerodynamics. The

motion of an airplane when the pilot changes the throttle setting of the engine is

described by the first law. The motion of a ball falling down through the atmosphere# or

a model rocket being launched up into the atmosphere are both examples of Newton)sfirst law. The motion of a kite when the wind changes can also be described by the first

law. 7e have created separate pages which describe each of these examples in more

detail to help you understand this important physical principle.

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Sir Isaac Newton first presented his three laws of motion in the &'rincipia (athematica

'hilosophiae Naturalis& in !"%". is second law defines a force to be eual to the

change in momentum with a change in time. (omentum is defined to be the mass m of 

an ob*ect times its velocity V.

8et us assume that we have an airplane at a point &0& defined by its location X0 and

time t0. The airplane has a mass m0and travels at velocity V0. The airplane is

sub*ected to an external force F and moves to a point &!&# which is described by a new

location X1 and time t1. The mass and velocity of the airplane change during the flight

to values m1 and V1. Newton)s second law can help us determine the new values

of V1 and m1# if we know how big the force F is. 8et us *ust take the difference between

the conditions at point &!& and the conditions at point &0&.

2 +m! 9! : m0 90, ; +t! : t0,

Newton)s second law talks about changes in momentum +m 9, so# at this point# we

can)t separate out how much the mass changed and how much the velocity changed.

7e only know how much product +m 9, changed.

8et us assume that the mass stays a constant value eual to m. This assumption is

pretty good for an airplane# the only change in mass would be for the fuel burned

between point &!& and point &0&. The weight of the fuel is probably small relative to theweight of the rest of the airplane# especially if we only look at small changes in time.. If

we were discussing the flight of a baseball# then certainly the mass remains a constant.

6ut if we were discussing the flight of a bottle rocket# then the mass does not remain a

constant and we can only look at changes in momentum. or a constant mass m#

Newton)s second law looks like1

2 m +9! : 90, ; +t! : t0,

The change in velocity divided by the change in time is the definition of the

acceleration a. The second law then reduces to the more familiar product of a mass andan acceleration1

2 m a

<emember that this relation is only good for ob*ects that have a constant mass. This

euation tells us that an ob*ect sub*ected to an external force will accelerate and that

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the amount of the acceleration is proportional to the si-e of the force. The amount of

acceleration is also inversely proportional to the mass of the ob*ect5 for eual forces# a

heavier ob*ect will experience less acceleration than a lighter ob*ect. /onsidering the

momentum euation# a force causes a change in velocity5 and likewise# a change in

velocity generates a force. The euation works both ways.

The velocity# force# acceleration# and momentum have both a magnitude and

a direction associated with them. Scientists and mathematicians call this a vector

uantity. The euations shown here are actually vector euations and can be applied in

each of the component directions. 7e have only looked at one direction# and# in

general# an ob*ect moves in all three directions +up:down# left:right# forward:back,.

The motion of an aircraft resulting from aerodynamic forces# aircraft weight#

and thrust can be computed by using the second law of motion.

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tr

Sir Isaac Newton first presented his three laws of motion in the &'rincipia (athematica

'hilosophiae Naturalis& in !"%". is third law states that for every action +force, in

nature there is an eual and opposite reaction. In other words# if ob*ect 4 exerts a forceon ob*ect 6# then ob*ect 6 also exerts an eual and opposite force on ob*ect 4. Notice

that the forces are exerted on different ob*ects.

or aircraft# the principal of action and reaction is very important. It helps to explain the

generation of lift from an airfoil. In this problem# the air is deflected downward by the

action of the airfoil# and in reaction the wing is pushed upward. Similarly# for a spinning

ball# the air is deflected to one side# and the ball reacts by moving in the opposite

direction. 4 jet engine also produces thrust through action and reaction. The engine

produces hot exhaust gases which flow out the back of the engine. In reaction# a

thrusting force is produced in the opposite direction.

Next to E = mc²# F = ma is the most famous euation in all of physics. =et many people remain mystified by this fairly

simple algebraic expression. It)s actually a mathematical representation of Isaac Newton)s second law of motion# one

of the great scientist)s most important contributions. The &second& implies that other laws exist# and# luckily for

students and trivia hounds everywhere# there are only two additional laws of motion. 4ll three are presented here#

using Newton)s own words1

!. >very ob*ect persists in its state of rest or uniform motion in a straight line unless it is compelled to

change that state by forces impressed on it.

$. orce is eual to the change in momentum per change in time. or a constant mass# force euals mass

times acceleration.

3. or every action# there is an eual and opposite reaction.These three laws form the foundation of what is known as classical mechanics# or the science concerned with the

motion of bodies being acted upon by forces. The bodies in motion could be large ob*ects# such as orbiting moons or

planets# or they could be ordinary ob*ects on >arth)s surface# such as moving vehicles or speeding bullets. >ven

bodies at rest are fair game.

Up e!t• "o# $saac e#ton

%or& ed

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• "o# the Enlightenment

%or&ed

• 'uriosity (roject)

*iggest +uestions ,aised -y

+uantum (hysics

7here classical mechanics begins to fall apart is when it tries to describe the motion of very small bodies# such as

electrons. 'hysicists had to create a new paradigm# known as uantum mechanics# to describe the behavior of

ob*ects at the atomic and subatomic level.

6ut uantum mechanics is beyond the scope of this article. ?ur focus will be classical mechanics and Newton)s three

laws. 7e)ll examine each in detail# both from a theoretical and a practical point of view. 7e)ll also discuss the history

of these laws# because how Newton arrived at his conclusions is *ust as important as the conclusions themselves.

The best place to start# of course# is at the beginning :: Newton)s first law.

e#ton/s econd a# 2a# of 3otion4

=ou may be surprised to learn that Newton wasn)t the genius behind the law of inertia. 6ut Newton himself wrote that

he was able to see so far only because he stood on &the shoulders of @iants.& 4nd see far he did. 4lthough the law of

inertia identified forces as the actions reuired to stop or start motion# it didn)t uantify those forces. Newton)s second

law supplied the missing link by relating force to acceleration. This is what it said1

7hen a force acts on an ob*ect# the ob*ect accelerates in the direction of the force. If the mass of an ob*ect is

held constant# increasing force will increase acceleration. If the force on an ob*ect remains constant#

increasing mass will decrease acceleration. In other words# force and acceleration are directly proportional#

while mass and acceleration are inversely proportional.

Technically# Newton euated force to the differential change in momentum per unit time. 3omentum is a

characteristic of a moving body determined by the product of the body)s mass and velocity. To determine the

differential change in momentum per unit time# Newton developed a new type of math :: differential calculus. is

original euation looked something like this1

F = 2m425v65t4

where the delta symbols signify change. 6ecause acceleration is defined as the instantaneous change in velocity inan instant of time +Av;At,# the euation is often rewritten as1

F = ma

The euation form of Newton)s second law allows us to specify a unit of measurement for force. 6ecause the

standard unit of mass is the kilogram +kg, and the standard unit of acceleration is meters per second suared +m;s$,#

the unit for force must be a product of the two :: +kg,+m;s$,. This is a little awkward# so scientists decided to use

a e#ton as the official unit of force. ?ne Newton# or N# is euivalent to ! kilogram:meter per second suared. There

are B.BB% N in ! pound.

So what can you do with Newton)s second lawC 4s it turns out# F = ma lets you uantify motion of every variety. 8et)s

say# for example# you want to calculate the acceleration of the dog sled shown below.

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$f you #ant to calculate the acceleration7 first you need to modify the force euation to get a = F6m. %hen you

plug in the num-ers for force 2100 4 and mass 280 &g47 you find that the acceleration is 9 m6s9.

 

Now let)s say that the mass of the sled stays at D0 kg and that another dog is added to the team. If we assume the

second dog pulls with the same force as the first +!00 N,# the total force would be $00 N and the acceleration would

be B m;s$.

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otice that dou-ling the force -y adding another dog dou-les the acceleration. :ppositely7 dou-ling the

mass to 100 &g #ould halve the acceleration to 9 m6s9.

inally# let)s imagine that a second dog team is attached to the sled so that it can pull in the opposite direction.

e#ton/s ;hird a# 2a# of Force (airs4

Newton)s third law is probably the most familiar. >veryone knows that every action has an eual and opposite

reaction# rightC Enfortunately# this statement lacks some necessary detail. This is a better way to say it1

 4 force is exerted by one ob*ect on another ob*ect. In other words# every force involves the interaction of two

ob*ects. 7hen one ob*ect exerts a force on a second ob*ect# the second ob*ect also exerts a force on the first

ob*ect. The two forces are eual in strength and oriented in opposite directions.

(any people have trouble visuali-ing this law because it)s not as intuitive. In fact# the best way to discuss the law of

force pairs is by presenting examples. 8et)s start by considering a swimmer facing the wall of a pool. If she places her 

feet on the wall and pushes hard# what happensC She shoots backward# away from the wall.

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TI(?T= 4. /84<=;4';@etty Images

;hat/s one hec& of a force<

/learly# the swimmer is applying a force to the wall# but her motion indicates that a force is being applied to her# too.

This force comes from the wall# and it)s eual in magnitude and opposite in direction.

Next# think about a book lying on a table. 7hat forces are acting on itC ?ne big force is >arth)s gravity. In fact# the

book)s weight is a measurement of >arth)s gravitational attraction. So# if we say the book weighs !0 N# what we)re

really saying is that >arth is applying a force of !0 N on the book. The force is directed straight down# toward the

center of the planet. Fespite this force# the book remains motionless# which can only mean one thing1 There must be

another force# eual to !0 N# pushing upward. That force is coming from the table.

If you)re catching on to Newton)s third law# you should have noticed another force pair described in the paragraph

above. >arth is applying a force on the book# so the book must be applying a force on >arth. Is that possibleC =es# it

is# but the book is so small that it cannot appreciably accelerate something as large as a planet.

=ou see something similar# although on a much smaller scale# when a baseball bat strikes a ball. There)s no doubt

the bat applies a force to the ball1 It accelerates rapidly after being struck. 6ut the ball must also be applying a force

to the bat. The mass of the ball# however# is small compared to the mass of the bat# which includes the batter

attached to the end of it. Still# if you)ve ever seen a wooden baseball bat break into pieces as it strikes a ball# then

you)ve seen firsthand evidence of the ball)s force.

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pplications and imitations of e#ton/s a#s

6y themselves# the three laws of motion are a crowning achievement# but Newton didn)t stop there. e took those

ideas and applied them to a problem that had stumped scientists for years :: the motion of planets. /opernicus

placed the sun at the center of a family of orbiting planets and moons# while the @erman astronomer Gohannes

Hepler proved that the shape of planetary orbits was elliptical# not circular. 6ut no one had been able to explain the

mechanics behind this motion. Then# as the story goes# Newton saw an apple fall to the ground and was sei-ed byinspiration. /ould a falling apple be related to a revolving planet or  moonC Newton believed so. This was his thought

process to prove it1

!. 4n apple falling to the ground must be under the influence of a force# according to his second law. That

force is gravity# which causes the apple to accelerate toward >arth)s center.

$. Newton reasoned that the moon might be under the influence of >arth)s gravity# as well# but he had to

explain why the moon didn)t fall into >arth. Enlike the falling apple# it moved parallel to >arth)s surface.

3. 7hat if# he wondered# the moon moved about the >arth in the same way as a stone whirled around at

the end of a stringC If the holder of the string let go :: and therefore stopped applying a force :: the stone

would obey the law of inertia and continue traveling in a straight line# like a tangent extending from the

circumference of the circle.

Stocktrek Images;@etty Images

>oes the moon move around the Earth in the same #ay that a stone #hirls around the end of astring?

B. 6ut if the holder of the string didn)t let go# the stone would travel in a circular path# like the face of a

clock. In one instant# the stone would be at !$ o)clock. In the next# it would be at 3 o)clock. 4 force is

reuired to pull the stone inward so it continues its circular path or orbit. The force comes from the holder

of the string.

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D. Next# Newton reasoned that the moon orbiting >arth was the same as the stone whirling around on its

string. >arth behaved as the holder of the string# exerting an inward:directed force on the moon. This

force was balanced by the moon)s inertia# which tried to keep the moon moving in a straight:line tangent

to the circular path.

". inally# Newton extended this line of reasoning to any of the planets revolving around the sun. >ach

planet has inertial motion balanced by a gravitational attraction coming from the center of the sun.

It was a stunning insight :: one that eventually led to the universal law of gravitation. 4ccording to this law# any twoob*ects in the universe attract each other with a force that depends on two things1 the masses of the interacting

ob*ects and the distance between them. (ore massive ob*ects have bigger gravitational attractions. Fistance

diminishes this attraction. Newton expressed this mathematically in this euation1

F = @2m1m9 6r 94

where F is the force of gravity between masses m1 and m9# @ is a universal constant and r  is the distance between

the centers of both masses.

?ver the years# scientists in *ust about every discipline have tested Newton)s laws of motion and found them to be

ama-ingly predictive and reliable. 6ut there are two instances where Newtonian physics break down. The first

involves ob*ects traveling at or near the speed of light. The second problem comes when Newton)s laws are applied

to very small ob*ects# such as atoms or subatomic particles that fall in the realm of uantum mechanics.

Still# these limitations shouldn)t take away from his accomplishments# so flip to the next page for more information

about Isaac Newton and other geniuses.

'rint

/itation Fate

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 4';@etty Images

-ase-all player shatters his -at.

These examples don)t show a practical application of Newton)s third law. Is there a way to put force pairs to good

useC Aet propulsion is one application. Esed by animals such as suid and octopi# as well as by certain airplanes

and rockets# *et propulsion involves forcing a substance through an opening at high speed. In suid and octopi# the

substance is seawater# which is sucked in through the mantle and e*ected through a siphon. 6ecause the animal

exerts a force on the water   *et# the water *et exerts a force on the animal# causing it to move. 4 similar principle is at

work in turbine:euipped *et planes and rockets in space.

Speaking of outer space# Newton)s other laws apply there# too. 6y using his laws to analy-e the motion of planets in

space# Newton was able to come up with a universal law of gravitation. 7e)ll explore this further in the next section.

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$f t#o dogs are on each side7 then the total force pulling to the left 2900 4 -alances the total force pulling to

the right 2900 4. ;hat means the net force on the sled is Bero7 so the sled doesnCt move.

This is important because Newton)s second law is concerned with net forces. 7e could rewrite the law to say1 7hen

a net force acts on an ob*ect# the ob*ect accelerates in the direction of the net force. Now imagine that one of the

dogs on the left breaks free and runs away. Suddenly# the force pulling to the right is larger than the force pulling to

the left# so the sled accelerates to the right.

7hat)s not so obvious in our examples is that the sled is also applying a force on the dogs. In other words# all forces

act in pairs. This is Newton)s third law :: and the topic of the next section.

'rint

/itation Fate

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e#ton/s First a# 2a# of $nertia4

May the Force Be with YouThe F, the m and the a in Newton's formula are veryimportant concepts in mechanics. The F isforce, a push or

 pull exerted on an object. The mis mass, a measure of how

much matter is in an object. And the a is acceleration, whichdescribes how an object's velocity changes over

time.Velocity, which is similar to speed, is the distance anobject travels in a certain amount of time.

8et)s restate Newton)s first law in everyday terms1

 4n ob*ect at rest will stay at rest# forever# as long as nothing pushes or pulls on it. 4n ob*ect in motion will stay

in motion# traveling in a straight line# forever# until something pushes or pulls on it.

The &forever& part is difficult to swallow sometimes. 6ut imagine that you have three ramps set up as shown below.

 4lso imagine that the ramps are infinitely long and infinitely smooth. =ou let a marble roll down the first ramp# which is

set at a slight incline. The marble speeds up on its way down the ramp. Now# you give a gentle push to the marble

going uphill on the second ramp. It slows down as it goes up. inally# you push a marble on a ramp that represents

the middle state between the first two :: in other words# a ramp that is perfectly hori-ontal. In this case# the marble will

neither slow down nor speed up. In fact# it should keep rolling. orever.

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ccording to e#ton/s first la#7 the mar-le on that -ottom ramp should just &eep going. nd going.

 

'hysicists use the term inertia to describe this tendency of an ob*ect to resist a change in its motion. The 8atin root

for inertia is the same root for &inert#& which means lacking the ability to move. So you can see how scientists came

up with the word. 7hat)s more ama-ing is that they came up with the concept. Inertia isn)t an immediately apparent

physical property# such as length or volume. It is# however# related to an ob*ect)s mass. To understand how# consider

the sumo wrestler  and the boy shown below.

 4';@etty Images

%hich person in this ring #ill -e harder to move? ;he sumo #restler or the little -oy?

8et)s say the wrestler on the left has a mass of !3" kilograms# and the boy on the right has a mass of 30 kilograms

+scientists measure mass in kilograms,. <emember the ob*ect of sumo wrestling is to move your opponent from his

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position. 7hich person in our example would be easier to moveC /ommon sense tells you that the boy would be

easier to move# or less resistant to inertia.

=ou experience inertia in a moving car  all the time. In fact# seatbelts exist in cars specifically to counteract the effects

of inertia. Imagine for a moment that a car at a test track is traveling at a speed of DD mph. Now imagine that a crash

test dummy is inside that car# riding in the front seat. If the car slams into a wall# the dummy flies forward into the

dashboard. 7hyC 6ecause# according to Newton)s first law# an ob*ect in motion will remain in motion until an outside

force acts on it. 7hen the car hits the wall# the dummy keeps moving in a straight line and at a constant speed until

the dashboard applies a force. Seatbelts hold dummies +and passengers, down# protecting them from their own

inertia.

Interestingly# Newton wasn)t the first scientist to come up with the law of inertia. That honor goes to @alileo and to

<enJ Fescartes. In fact# the marble:and:ramp thought experiment described previously is credited to @alileo. Newton

owed much to events and people who preceded him. 6efore we continue with his other two laws# let)s review some of 

the important history that informed them.

'rint

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