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.
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.
© 2013 Pearson Education, Inc.
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
© 2013 Pearson Education, Inc.
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
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