Physics 40 1-D Kinematics

Physics 40 IS Classical Mechanics!

Study of the motion of objects and mechanical systems that are large

relative to atoms and move at speeds much slower than the speed of light.

Isaac Newton (1642 -1727)

In Principia (1687 ) Newton •Invented Calculus •3 Laws of Motion •Universal Law of Gravity The force of gravity is Universal: The same force that makes

an apple fall to Earth, causes the moon to fall around the Earth and the planets to orbit the Sun.

Our Goal….

Celestial Mechanics!

Let no one unversed in geometry enter here. The Universe is made of pure mathematical ideas – the Platonic

Solids. Plato believed that the stars, planets, Sun and Moon move round the Earth in crystalline spheres.

Earth and the universe were seen as constructed out of five basic elements: earth, water, air, fire, and ether. The natural place of the motionless Earth was at the center of that universe. The stars in the heavens were made up of an indestructible substance called ether (aether) and were considered as eternal and unchanging. The laws of nature of the Earth were different from those of the Heavens.

Naïve Science: From our perspective, the sun and stars appear to orbit us!

Ptolemy's Geocentric Model

of the Universe 150 AD

Problem with the Theory: Apparent Retrograde Motion of Planets In a Geocentric Model there shouldn’t be

Retrograde motion.

Ptolemy 85-165 AD “Saving the Appearances”

The Sun and the planets would revolve in small circles whose centers revolve in large circles about the Earth ("epicycles"). 150 AD

As Christianity started taking over the Roman Empire, Paganism was illegal including astronomy. The Burning of the Library at Alexandria in 391 AD

destroyed scientific texts. The murder of Hypatia marks the end of the Golden Age of the Greeks and

the dawn of the European Dark Ages…..

An avowed paganist in a time of religious strife, Hypatia was also one of the first women to study math, astronomy and

philosophy. ne day on the streets of Alexandria, Egypt, in the year 415 or 416, a mob of Christian zealots led by Peter the Lector accosted a woman’s carriage and dragged her from it

and into a church, where they stripped her and beat her to death with roofing tiles. They then tore her body apart and burned it

5th-15th Centuries

Developed science & medicine based on observation and

experiment, rather than on conjecture creating the basis of

what would later be called The Scientific Method.

Recovery of Aristotle spanned about 100 years, from the middle 12th century into the 13th century, and copied or translated over 42 books from Arabic texts into latin. Aristotle's newly translated views discounted the notions of a personal God, immortal soul, or creation which was counter to church dogma. His books included physics and astronomy. Galileo read Aristotle and then challenged his ideas, using the scientific method of experimentation invented by Islamic scientists. Hence began the start of modern physics & the Renaissance. Without Islamic scientists keeping science alive during the dark ages, Europe might still be in the dark ages!

European Enlightenment Renaissance

14th & 15th Century

The Vitruvian Man 1490

De revolutionibus orbium coelestium On the Revolutions of the Heavenly Spheres, 1543 If the Sun is at the Center of the Solar System you

don’t need epicycles.

REVOLUTION!

Catholic Inquisition The Catholic Congregation for the Doctrine of the Faith, ruled that the propositions that the Sun is immobile and at the center of the universe and that the Earth moves around it, are both "foolish and absurd in philosophy," and the first to be "formally heretical" and the second "at least erroneous in faith" in theology.

The Rejection of the Copernican Heliocentric Model: No Stellar Parallax

I hold that the Sun is located at the centre of the revolutions of the heavenly orbs and does not change place, and that the Earth rotates on itself and moves

around it.

Heliocentric Heretics

Rome, Campo de'fiori: The monument to Giordano Bruno, burnt at the stake here.

The Trial of Galileo

June 22, 1633: Galileo was convicted and sentenced to life imprisonment by the Catholic Inquisition.

In 1992, the church finally lifted its edict of Inquisition against Galileo, who went to his grave a devout Catholic, despite the

church’s treatment of him.

Tycho Brahe and Johanes Kepler

Tycho was a great observational astronomer and took detailed data of planetary motion. Kepler worked for Tycho as his mathematician. Kepler introduced physics into astronomy for the first time and derived his laws of planetary motion from Tycho’s observational data. Kepler’s Laws are thus empirical - based on observation and not theory.

Kepler’s 3 Laws of Planetary Motion

1: The orbit of each planet about the sun is an ellipse with the sun at one focus.

2. Each planet moves so that it sweeps out equal areas in equal times.

213

1

constantTr

=

3. The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit.

Based on observational data he derived three laws of planetary motion that put the sun at he center of the

universe with elliptical orbits.

"The next question was - what makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around an orbit. As you will see, the answer is not very far from the truth. The only difference is that the angels sit in a different direction and their wings push inward." -Richard Feynman

Isaac Newton (1642 -1727)

In Principia (1687 ) Newton •Invented Calculus •3 Laws of Motion •Universal Law of Gravity

Using his Calculus, Newton derives Kepler’s three laws of planetary motion from his own

three laws of motion and his Universal Law of Gravity. Newton is the man of the millennium.

325 years later we know…..

NOT ONLY is the Earth not immobile!

The Earth Moves through Solar System at 30Km/sec!!!

464m/s

Precession causes the position of the

North Pole to change over a

period of 26,000 years.

Orbital Speed of Earth: ~ 30 km/s

Milky Way Galaxy

Orbital Speed of Solar System: 220 km/s Orbital Period: 225 Million Years

Universe expands with Hubble Flow….

Translational Motion Circular Motion

Projectile Motion Rotational Motion

Types of Motion

Natural Motion •Objects have a proper place •Objects seek their natural place •External forces must be constantly applied to moving objects in order to keep them going. •The heavier the object, the faster it falls. •Did not experiment to test theories.

Galileo Challenged The Dogma Of Natural Motion with

Experiments

The natural motion of a body is to remain in

whatever state of motion it is in unless

acted upon by net external forces.

Galileo Challenged The Dogma Of Natural Motion

Galileo Challenged Aristotle Physics In a vacuum, all objects fall with the same

acceleration due to gravity: 9.80 m/s2, independent of their weight.

Galileo’s Motion Studies

0

2

f

xvt

v vv

vat

∆=∆+

=

∆=∆

gave us…

Definitions:

Distance and Displacement

(delta) means "change in" = 'final - initial'∆

∆

The total distance traveled relative to an origin. Distance is a scalar.

Displacement is a vector. The unit is the meter.

0fx x x∆ = −

An ant zig-zags back and forth on a picnic table as shown. The ant’s distance traveled and displacement are A. 50 cm and 50 cm. B. 30 cm and 50 cm. C. 50 cm and 30 cm. D. 50 cm and –50 cm. E. 50 cm and –30 cm.

QuickCheck 2.1

Slide 2-29

An ant zig-zags back and forth on a picnic table as shown. The ant’s distance traveled and displacement are A. 50 cm and 50 cm. B. 30 cm and 50 cm. C. 50 cm and 30 cm. D. 50 cm and –50 cm. E. 50 cm and –30 cm.

QuickCheck 2.1

Slide 2-30

Average Speed &Velocity Speed is how fast something moves.

The average speed is the total distance per time. The average velocity is the the total displacement per time.

Velocity is a vector. The unit is m/s.

total displacementtotal time

xvt

∆= =

∆

Sense of Speed

1 / 3.6 / 2.24 /10 / 36 / 22.4 /20 / 72 / 44.8 /30 / 108 / 67.2 /

m s km hr mi hrm s km hr mi hrm s km hr mi hrm s km hr mi hr

= == == == =

1 / 2.25 /m s mi hr≈

1 / 0.62 /km hr mi hr≈

Acceleration How fast How fast is changing.

The rate at which the speed is changing.

Speeding up

Slowing down

Constant speed, changing direction.

change in velocitychange in time

vat

∆= =

∆

Acceleration is in the direction of the net Force but not necessarily

in the direction of velocity. Velocity is always in the direction of the motion!

Galileo’s Motion Studies

0 , , 2

fv vx vv v at t

+∆ ∆= = =∆ ∆

gave us…

Kinematic Equations

With a little al-jbr….

0 , , 2

fv vx vv v at t

+∆ ∆= = =∆ ∆

vat

∆=∆

Start:

0fv v a t→ − = ∆

Assume constant acceleration!

0fv v a t= + ∆

0 , , 2

fv vx vv v at t

+∆ ∆= = =∆ ∆

vat

∆=∆

Start:

0fv v a t→ − = ∆

Assume constant acceleration!

0fv v a t= + ∆

0 , , 2

fv vx vv v at t

+∆ ∆= = =∆ ∆

0

2fv vx

t+∆

=∆

Start:

0 0( )2

f ix x v v a tt− + + ∆

→ =∆

20

12f ix x v t a t= + ∆ + ∆

0 , , 2

fv vx vv v at t

+∆ ∆= = =∆ ∆

0

2fv vx

t+∆

=∆ 0

2

f

t xv v

→∆ = ∆+

vat

∆=∆

Start:

Combine & Eliminate t:

0 = fv vt

a−

→ ∆

2 0

0

v vft x

v v af

−∆ = ∆ =

+

2 20 2fv v a x= + ∆Algebra:

Galileo’s Motion Studies

0

0

20

2 20

, , 2

1 22

f

f

f

v vx vv v at t

v v at

x v t at

v v a x

+∆ ∆= = =∆ ∆

= +

∆ = +

= + ∆

gave us…

Kinematic Equations

Acceleration: Changing Velocity From t = 0, how long does it take the car to come to a full stop?

How far does the car travel before it comes to a stop?

+x

Acceleration: Changing Velocity

2

Knowns5 /

28 /0

?

i

f

a m sv m svt

= −==

=

f iv vt

a−

=

2

0 28 / 5.65 /

m s sm s

−= =

−

f iv v at= +Which equation to use?

Solve for t:

5.6t s=

Acceleration: Changing Velocity

From t = 0, to t = 5.6s, how far does the car travel before it comes to a stop?

+x

2

Knowns5 /

28 /0

5.6

i

f

a m sv m svt s

= −==

=

Which equation? 20

12

x v t at∆ = +

2 2128 5.6 ( 5 / )(5.6 ) 78.42

mx s m s s ms

∆ = + − =

78.4x m∆ =

YOU TRY IT!

Galileo’s Motion Studies

0

0

20

2 20

, , 2

1 22

f

f

f

v vx vv v at t

v v at

x v t at

v v a x

+∆ ∆= = =∆ ∆

= +

∆ = +

= + ∆

gave us…

Definitions of averages

Kinematic Equations with constant acceleration

Constant vs Changing Acceleration Depends on the FORCE

Constant Forces (Use Kinematic Equations)

• Constant pushes and pulls • Inclined planes • Gravity near the earth (Free Fall) • Pulleys, Conical Pendulums

Variable Forces (Use Calculus!)

• Springs and Pendulums • Air Resistance • Gravity Far from Earth • Electricity and Magnetism • MOST FORCES!!!!

Newton’s Calculus will give us INSTANTEOUS motion…

© 2013 Pearson Education, Inc.

Even if the velocity is not constant, we can divide the motion into N steps in which it is approximately constant, and compute the final position as:

The integral may be interpreted graphically as the total area enclosed between the t-axis and the velocity curve.

The total displacement ∆s is called the “area under the curve.”

Finding Position From Velocity

Slide 2-55

1-D Motion in a nutshell 0

2

2

Averages: , , 2

Instantaneous: , ,

fv vx vv v at t

dx dv d xv a adt dt dt

+∆ ∆= = =∆ ∆

= = =

fv fv

0

20 0

2 20

Kinematics Eqs:

1 2

2

f

f

f

v v at

x x v t at

v v a x

= +

= + +

= + ∆

Constant acceleration.

00

( )t

fv v a t dt= + ∫

00

( )t

fx x v t dt= + ∫

Varying acceleration.

Motion Graphs

What kind of motion is this?

What kind of motion is this?

xvt

∆=∆

3400 1 /400

mv m ss

−= = −2 0 /v m s=

1400 2 /200

mv m ss

= =

What is the velocity during each segment?

MORE Motion Graphs

Is the acceleration constant or changing during the motion? Find the equation for the displacement.

A-B: B-2s: 2s-D:

Object moves backwards with average speed of 2m/s, slows down and stops.

Object moves forward with average speed of 2m/s, speeding up until it comes back to where it started.

Object continues to move forward and increasing speed.

What kind of motion does this graph represent? What is the NARRATIVE? (story)

2( ) 4 2x t t t= − +

Motion Graphs What is the average velocity between A and B?

xxvt

∆=∆

2 01m

s− −

=

(1 ) (0 )(1 0 )

x s x ss s−

=−

2 ms

= −

Motion Graphs What is the average velocity between B and D?

xxvt

∆=∆

6 ( 2 )2− −

=m m

s

(3 ) (1 )(3 1 )

−=

−x s x s

s s

4=ms

Instantaneous Velocity The velocity at any time t is the slope of the x vs t graph at t.

( )xdxv tdt

=

2( 4 2 )( ) 4 4d t tv t tdt

− += = − +

2(2.5 ) 4 4 (2.5 ) 6m m mv s ss s s

= − + =

What is the instantaneous velocity at t=2.5s?

What does the velocity vs time graph look like?

2( ) 4 2x t t t= − +

Velocity Graph

2( ) 4 2x t t t= − +

What does the a-t graph look like?

All the Graphs

2( ) 4 2x t t t= − +

24 /xa m s=

24 /a m s=( ) 4 4v t t= − +

What is the displacement from zero to 2s?

m/s

(s)

In general……

0

( )t

xx t v dt= ∫

Displacement = area under the v-t graph

0

( )t

xa t dt= ∫

1 ( )2 xa t t= 1 (base)(height)

2=

Area under graph=

212 xa t=

What is the displacement from zero to 2s?

m/s

(s)

1 (base)(height)2

x =

1 1(1 )(-4 / ) (1 )(4 / ) 02 2

= + =s m s s m s

What is the displacement from zero to 2s? 2( ) 4 2x t t t= − +

Displacement = area under the v-t graph

(2 ) 0x s =

1 (base)(height)2

x =

1 1(1s)(-4m/s)+ (1s)(4m/s)=02 2

=

m/s

(s)

Zero!

What is the displacement from zero to 4s? 2( ) 4 2x t t t= − +

Displacement = area under the v-t graph

(4 ) 16x s m=

1 (base)(height)2

x =

1 1(1s)(-4m/s)+ (3s)(12m/s)=16m2 2

=

m/s

(s)

Which velocity-versus-time graph goes with the position-versus-time graph on the left?

Which velocity-versus-time graph goes with the position-versus-time graph on the left?

Deriving Graphs from Graphs

Derive x-t and a-t graphs and find displacement equations for each segment using equations of lines and integration. Assume x(0)=0.

Free Fall Unless told otherwise, ignore air resistance for

free fall problems!

Galileo Challenged Aristotle Physics In a vacuum, all objects fall with the same

acceleration due to gravity: 9.80 m/s2, independent of their weight.

Acceleration of Freely Falling Object

• The acceleration of an object in free fall is directed downward, regardless of the initial motion

• The magnitude of free fall acceleration is g = 9.80 m/s2

g decreases with increasing altitude – g varies with latitude – 9.80 m/s2 is the average at the Earth’s

surface – We will neglect air resistance – g is a SCALAR!!! POSTIVE

Free Fall Equations For any object in the absence of air resistance.

29.80 /ya g m s= − = −

0

20

2 20

Customize:

12

2

f

f

v v gt

y v t gt

v v g y

= −

∆ = −

= − ∆

0

20

2 20

Kinematic Eqs:

1 22

f

f

v v at

x v t at

v v a x

= +

∆ = +

= + ∆Note: v0 can be negative!

(taking up as +y)

Falling from Rest

2 21 52

y at t∆ = =

2

:~ 10 /

Estimatea g m s=

10v at t= =

20 /20

v m sy m=

∆ =

10 /5

v m sy m=

∆ =

30 /45

v m sy m=

∆ =

40 /80

v m sy m=

∆ =

50 /125

v m sy m=

∆ =

+

0

20

12

fv v gt

y v t gt

= +

∆ = +

0 0v =

!v y≠ ∆How FAR is not How FAST!

Take down as +y:

How Far: y(t) ~ t2

0fv v at= +

20

12

y v t at∆ = +

How Fast: v(t) ~ t1

+

How Fast How Fast is Changing! 29.80 /g m s=

FIRST: Define Reference Frame In this reference frame,what is the sign of a? 29.80 /a m s= −

What is v at t = 3s?

0fv v at= +

20 9.80 (3 )m ss

= −

29.4 ms

= −

20: 0, 9.8 / , 3Knowns v a m s t s= = − =

: ?fUnknown v =

Negative because it is moving downward, in the negative direction!

FIRST: Define Reference Frame

20: 0, 9.8 / , 3 , 29.4 /fKnowns v a m s t s v m s= = − = = −

: ?Unknown y∆ =

The displacement is negative because it is moves downward, in the negative direction but “how far” is a distance – a scalar – and is positive!

How far did the ball fall in those 3 seconds?

20

1

2y v t at∆ = +

22

1

20 ( 9.8 )(3 )m s

s= + −

44.1m= −

The ball fell 44.1m.

Throwing up is Also Free Fall! Symmetry of G Field.

2

:~ 10 /

Estimatea g m s=

0

20

12

fv v gt

y v t gt

= +

∆ = +

What Goes Up Must Come Down Someone standing at the edge of a cliff throws one ball straight up and one straight down at the same speed. Ignoring air resistance, which ball strikes the ground with the greatest speed?

Free Fall Question: You throw the rock down with an initial speed of 30 m/s. The rock hits the ground in 3 seconds. With what speed will the rock hit the ground?

+y

230 9.8 (3 )m m ss s

= − −

59.4fmvs

= −

0fv v at= +

How high is the cliff?

20: 30 / , 9.8 / , 3Knowns v m s a m s t s= − = − =

: ?fUnknown v =

Free Fall

20

1

2y v t at∆ = +

2 21

2( 30 / )(3 ) ( 9.8 / )(3 )m s s m s s= − + −

134m= −

The cliff is 134 m high.

20: 30 / , 9.8 / , 3Knowns v m s a m s t s= − = − =

: ?Unknown y∆ =+y

Question: You throw the rock down with an initial speed of 30 m/s. The rock hits the ground in 3 seconds. With what speed will the rock hit the ground? How high is the cliff?

Free Fall: Throwing Up What is the speed at the top of the path? ZERO! What is the acceleration at the top? a = -9.80 m/s2 What is the velocity at the same height on the way down? -30 m/s

+y

With what velocity will the rock hit the ground? -59.4 m/s SAME as if you threw it straight down at 30m/s!

How long does it take to hit the ground? First try to guess!

+y 0fv v at= +

02

59.4 / 30 /9.8 /

fv v m s m sta m s− − −

= =−

9.12t s=

20: 30 / , 9.8 / , 3 , 59.4 /fKnowns v m s a m s t s v m s= = − = = −

: ?Unknown t∆ =

How long to the top? How long back to launch point? Final v increases by 30m/s?

I guess about 9 seconds!

Free Fall: Throwing Up Problem

A ball is tossed straight up in the air. At its very highest point, the ball’s instantaneous acceleration ay is A. Positive. B. Negative. C. Zero.

QuickCheck 2.18

Slide 2-96

A ball is tossed straight up in the air. At its very highest point, the ball’s instantaneous acceleration ay is A. Positive. B. Negative. C. Zero.

QuickCheck 2.18

Slide 2-97

Stopping Distance

Traveling at 70 miles per hour, what is your breaking distance?

If v doubles, d quadruples!!!

20

2vx

a∆ = −

Stopping Distance goes as the SQUARE of the speed!

2 20 2fv v a x= + ∆

Brake Question You are driving a car going 80 km/hr (50mph) when a head-on collision happens 25 meters ahead of you. If you can brake at 6 m/s2, will you hit the crash or stop before it?

2 20 2fv v a x= + ∆

20

2vx

a∆ = −

2

2

(22 / ) 40.3 252( 6 / )

m sx m mm s

∆ = − = >−

20: 80 / 22 / , 0, 6 / fKnowns v km hr m s v a m s= = = = −

: ?Unknown x∆ =

CRASH!

Stopping Distance goes as the SQUARE of the speed!

HW: Speedy Sally Speedy Sally, driving at 30.0 m/s, enters a one-lane tunnel. She then observes a slow-moving van 155 m ahead traveling at 5.00 m/s. Sue applies her brakes but can accelerate only at −2.00 m/s2 because the road is wet. Will there be a collision? If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sally's car and the van. Sketch the x-t graphs for both the vehicles. What does it mean?

HW: Rock Drop

A rock is dropped from rest into a well. The sound of the splash is heard 3.20 s after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s.

Rank in order, from largest to smallest, the accelerations aA– aC at points A – C.

A) aA > aB > aC B) aA > aC > aB C) aB > aA > aC D) aC > aA > aB E) aC > aB > aA

A) aA > aB > aC B) aA > aC > aB C) aB > aA > aC D) aC > aA > aB E) aC > aB > aA

Rank in order, from largest to smallest, the accelerations aA– aC at points A – C.

Here is a motion diagram of a car speeding up on a straight road: The sign of the acceleration ax is A. Positive. B. Negative. C. Zero.

QuickCheck 2.13

Slide 2-69

Here is a motion diagram of a car speeding up on a straight road: The sign of the acceleration ax is A. Positive. B. Negative. C. Zero.

QuickCheck 2.13

Slide 2-70

Speeding up means vx and ax have the same sign.

Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram?

QuickCheck 2.5

Slide 2-44

Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram?

QuickCheck 2.5

Slide 2-45

Here is a position graph of an object: At t = 3.0 s, the object’s velocity is A. 40 m/s. B. 20 m/s. C. 10 m/s. D. –10 m/s. E. None of the above.

QuickCheck 2.7

Slide 2-50

Here is a position graph of an object: At t = 3.0 s, the object’s velocity is A. 40 m/s. B. 20 m/s. C. 10 m/s. D. –10 m/s. E. None of the above.

QuickCheck 2.7

Slide 2-51

Which velocity-versus-time graph or graphs goes with this acceleration-versus-time graph? The particle is initially moving to the right and eventually to the left.

Which velocity-versus-time graph or graphs goes with this acceleration-versus-time graph? The particle is initially moving to the right and eventually to the left.

Which position-versus-time graph goes with the velocity-versus-time graph at the top? The particle’s position at ti = 0 s is xi = –10 m.

Which position-versus-time graph goes with the velocity-versus-time graph at the top? The particle’s position at ti = 0 s is xi = –10 m.