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.

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

xv

t

v vv

va

t

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

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 displacement

total time

xv

t

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 velocity

change in time

va

t

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 a

t t

gave us…

Kinematic Equations

With a little al-jbr….

0 , ,

2

fv vx vv v a

t t

va

t

Start:

0fv v a t

Assume constant acceleration!

0fv v a t

0 , ,

2

fv vx vv v a

t t

va

t

Start:

0fv v a t

Assume constant acceleration!

0fv v a t

0 , ,

2

fv vx vv v a

t t

0

2

fv vx

t

Start:

0 0( )

2

f ix x v v a t

t

2

0

1

2f ix x v t a t

0 , ,

2

fv vx vv v a

t t

0

2

fv vx

t

0

2

f

t xv v

va

t

Start:

Combine &

Eliminate t:

0 =

fv vt

a

2 0

0

v vf

t xv v a

f

2 2

0 2fv v a x Algebra:

Galileo’s Motion Studies

0

0

2

0

2 2

0

, , 2

1

2

2

f

f

f

v vx vv v a

t t

v v at

x v t at

v v a x

gave us…

Kinematic Equations

Quicky Question

An automobile enters a freeway on-ramp at 15.0m/s and accelerates

uniformly up to 25.0 m/s in a time of 10.0s.

a) What is the automobile’s average velocity?

0

2

f

ave

v vv

15 / 25 /20 /

2

m s m sm s

Which equation?

An automobile enters a freeway on-ramp at 15.0m/s and accelerates

uniformly up to 25.0 m/s in a time of 10.0s.

b) What is the automobile’s average acceleration?

f iv vva

t t

225 / 15 /1 /

10

m s m sm s

s

Which equation?

Quicky Question

An automobile enters a freeway on-ramp at 15.0m/s and accelerates

uniformly up to 25.0 m/s in a time of 10.0s.

c) What is the distance traveled in this amount of time?

Which equation?

Quicky Question

2

0

1

2x v t at 2

2

1

215 / (10 ) 1 (10 )

mm s s s

s

200x m

21 /a m s

(you could also use vave equation.)

Motion Diagrams

Skiing through the woods

Draw the Motion Diagram

An automobile enters a freeway on-ramp at 15.0m/s and accelerates

uniformly up to 25.0 m/s in a time of 10.0s.

Problem Solving Strategy

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

Draw the Motion Graph

Acceleration: Changing Velocity

2

Knowns

5 /

28 /

0

?

i

f

a m s

v m s

v

t

f iv vt

a

2

0 28 /5.6

5 /

m ss

m 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

Knowns

5 /

28 /

0

5.6

i

f

a m s

v m s

v

t s

Which equation? 2

0

1

2x v t at

2 2128 5.6 ( 5 / )(5.6 ) 78.4

2

mx s m s s m

s

78.4x m

YOU TRY IT!

Motion Graphs

What kind of motion is this?

What kind of motion is this?

xv

t

3

4001 /

400

mv m s

s

2 0 /v m s

1

4002 /

200

mv m s

s

What is the velocity during each

segment?

Which position-versus-time graph represents the motion shown in the

motion diagram?

Which position-versus-time graph represents the motion shown in the motion diagram?

The ball rolls up the ramp, then back down. Which is the correct acceleration graph?

The ball rolls up the ramp, then back down. Which is the correct acceleration graph?

Motion Graphs

What is the average velocity between A and B?

x

xv

t

2 0

1

m

s

(1 ) (0 )

(1 0 )

x s x s

s s

2m

s

Motion Graphs

What is the average velocity between B and D?

x

xv

t

6 ( 2 )

2

m m

s

(3 ) (1 )

(3 1 )

x s x s

s s

4m

s

Rank in order, from largest to smallest, the accelerations a

A– a

C at

points A – C.

A) aA > a

B > a

C

B) aA

> aC > a

B

C) aB

> aA > a

C

D) a

C > a

A > a

B

E) aC > aB > aA

A) aA > a

B > a

C

B) aA

> aC > a

B

C) aB

> aA > a

C

D) a

C > a

A > a

B

E) aC > aB > aA

Rank in order, from largest to smallest, the accelerations a

A– a

C at

points A – C.

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

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.

Galileo’s Motion Studies

0

0

2

0

2 2

0

, , 2

1

2

2

f

f

f

v vx vv v a

t t

v v at

x v t at

v v a x

gave us…

Definitions of averages

Kinematic Equations with constant acceleration

Newton’s Calculus

will give us INSTANTEOUS motion…

Newton’s Calculus

2

2

( )( )

( ) ( )( )

dx tv t

dt

dv t d x ta t

dt dt

will give us INSTANTEOUS motion…

0

( ) ( )

t

v t a t dt 0

( )

t

xx t v dt

1-D Motion in a nutshell

0

2

2

Averages: , , 2

Instantaneous: , ,

fv vx vv v a

t t

dx dv d xv a a

dt dt dt

fv fv

0

2

0 0

2 2

0

Kinematics Eqs:

1

2

2

f

f

f

v v at

x x v t at

v v a x

Constant acceleration.

0

( )

t

f fv v a t dt

0

0

( )

t

fx x v t dt

Varying acceleration.

Constant vs Changing Acceleration

Depends on the FORCE

Constant Forces

• Constant pushes and pulls

• Inclined planes

• Gravity near the earth (Free Fall)

• Pulleys, Conical Pendulums

Variable Forces

• Springs and Pulleys

• Air Resistance

• Gravity Far from Earth

• Electricity and Magnetism

• MOST FORCES!!!!

© 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

• Figure (a) shows a realistic

velocity-versus-time graph

for a car leaving a stop sign.

• The graph is not a straight line,

so this is not motion with a

constant acceleration.

• Figure (b) shows the car’s

acceleration graph.

• The area under the curve is the change

in velocity:

Slide 2-112

Instantaneous Acceleration

Slide 2-114

Motion Graphs

Is the acceleration constant or changing during the motion?

Find the equation for the displacement.

A-B:

B-C:

C-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

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

( )x

dxv t

dt

2( 4 2 )( ) 4 4

d t tv t t

dt

2(2.5 ) 4 4 (2.5 ) 6

m m mv s s

s 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( )

2xa t t

1(base)(height)

2

Area under graph

21

2xa t

What is the displacement from zero to 2s?

m/s

(s)

1(base)(height)

2x

1 1(1 )(-4 / ) (1 )(4 / ) 0

2 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)

2x

1 1(1s)(-4m/s)+ (1s)(4m/s)=0

2 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)

2x

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

2 2

m/s

(s)

1-D Motion in a nutshell

0

2

2

Averages: , , 2

Instantaneous: , ,

fv vx vv v a

t t

dx dv d xv a a

dt dt dt

fv fv

0

2

0 0

2 2

0

Kinematics Eqs:

1

2

2

f

f

f

v v at

x x v t at

v v a x

Constant acceleration.

0

( )

t

f fv v a t dt

0

0

( )

t

fx x v t dt

Varying acceleration.

Last Time…..

2( ) 4 2x t t t

24 /xa m s

24 /a m s( ) 4 4v t t

Deriving Graphs from Graphs

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

2

0

2 2

0

Customize:

1

2

2

f

f

v v gt

y v t gt

v v g y

0

2

0

2 2

0

Kinematic Eqs:

1

2

2

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

2 215

2y at t

2

:

~ 10 /

Estimate

a g m s

10v at t

20 /

20

v m s

y m

10 /

5

v m s

y m

30 /

45

v m s

y m

40 /

80

v m s

y m

50 /

125

v m s

y m

+

0

2

0

1

2

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

2

0

1

2y 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 )

ms

s

29.4m

s

2

0: 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

2

0: 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?

2

0

1

2y v t at

2

2

1

20 ( 9.8 )(3 )

ms

s

44.1m

The ball fell 44.1m.

Throwing up is Also Free Fall!

Symmetry of G Field.

2

:

~ 10 /

Estimate

a g m s

0

2

0

1

2

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 ms

s s

59.4f

mv

s

0fv v at

How high is the cliff?

2

0: 30 / , 9.8 / , 3Knowns v m s a m s t s

: ?fUnknown v

Free Fall

2

0

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.

2

0: 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

0

2

59.4 / 30 /

9.8 /

fv v m s m st

a m s

9.12t s

2

0: 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!!!

2

0

2

vx

a

Stopping

Distance

goes as the

SQUARE

of the

speed!

2 2

0 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 2

0 2fv v a x 2

0

2

vx

a

2

2

(22 / )40.3 25

2( 6 / )

m sx m m

m s

2

0: 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.

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Figure (a) shows the motion diagram of an object sliding down a straight, frictionless inclined plane.

Figure (b) shows the the free-fall acceleration the object would have if the incline suddenly vanished.

This vector can be broken into two pieces: and .

The surface somehow “blocks” , so the one-dimensional acceleration along the incline is

The correct sign depends on the direction the ramp is tilted.

Motion on an Inclined Plane

Slide 2-102

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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 hr

m s km hr mi hr

m s km hr mi hr

m s km hr mi hr

1 / 2.25 /m s mi hr

1 / 0.62 /km hr mi hr