Physics 40 1-D Kinematics

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1 Physics 40 1-D Kinematics

2 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.

3 Isaac Newton ( ) 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.

4 Our Goal. Celestial Mechanics!

6 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.

The stars in the heavens were made up of an indestructible substance called ether (aether) and were

7 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.

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

9 Ptolemy's Geocentric Model of the Universe 150 AD

10 Problem with the Theory: Apparent Retrograde Motion of Planets In a Geocentric Model there shouldn t be Retrograde motion.

11 Ptolemy 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

The Burning of the Library at Alexandria in 391 AD destroyed

12 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..

13 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

14 5 th -15 th Centuries

18 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 1th century into the 13th century, and copied or translated over 4 books from Arabic texts into latin.

24 Recovery of Aristotle spanned about 100 years, from the middle 1th century into the 13th century, and copied or translated over 4 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!

25 European Enlightenment Renaissance 14 th & 15 th Century The Vitruvian Man 1490

26 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.

27 REVOLUTION!

28 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. Catholic Inquisition

29 The Rejection of the Copernican Heliocentric Model: No Stellar Parallax

30 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.

31 Heliocentric Heretics

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

33 The Trial of Galileo June, 1633: Galileo was convicted and sentenced to life imprisonment by the Catholic Inquisition. In 199, 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.

34 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.

35 Based on observational data he derived three laws of planetary motion that put the sun at he center of the universe with elliptical orbits.

36 "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 (164-177) Using his Calculus, Newton derives Kepler s three laws of planetary motion from his own three laws of motion and his

37 Isaac Newton ( ) 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. In Principia (1687 ) Newton Invented Calculus 3 Laws of Motion Universal Law of Gravity

38 35 years later we know..

39 NOT ONLY is the Earth not immobile! The Earth Moves through Solar System at 30Km/sec!!!

40 464m/s

41 Precession causes the position of the North Pole to change over a period of 6,000 years.

42 Orbital Speed of Earth: ~ 30 km/s

43 Milky Way Galaxy Orbital Speed of Solar System: 0 km/s Orbital Period: 5 Million Years

44 Universe expands with Hubble Flow.

45 Types of Motion Translational Motion Circular Motion Projectile Motion Rotational Motion

46 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.

47 Galileo Challenged The Dogma Of Natural Motion with Experiments

48 Galileo Challenged The Dogma Of Natural Motion The natural motion of a body is to remain in whatever state of motion it is in unless acted upon by net external forces.

49 Galileo Challenged Aristotle Physics In a vacuum, all objects fall with the same acceleration due to gravity: 9.80 m/s, independent of their weight.

50 Galileo s Motion Studies gave us Definitions: x v t v0 v v v a t f

51 Distance and Displacement The total distance traveled relative to an origin. Distance is a scalar. Displacement is a vector. The unit is the meter. x x x f 0 (delta) means "change in" = 'final - initial'

52 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. v total displacement total time x t

53 Acceleration How fast How fast is changing. The rate at which the speed is changing. a change in velocity change in time v t Speeding up Slowing down Constant speed, changing direction.

54 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!

55 Galileo s Motion Studies gave us Kinematic Equations x v v f t 0 v, v, a v t With a little al-jbr.

56 Start: x v v f t 0 v, v, a v t Assume constant acceleration! v a t v v a t f 0 v v a t f 0

57 Start: x v v f t 0 v, v, a v t Assume constant acceleration! v a t v v a t f 0 v v a t f 0

58 Start: v0 v f x v, v, a t v t x t v0 v f xf xi v0 ( v0 a t) t 1 x f xi v0 t a t

59 Start: v0 v f x v, v, a t v t v0 v f x t t v0 a v t t v = f v v a f 0 x Combine & Eliminate t: t v v f 0 x v v a 0 f Algebra: f 0 v v a x

60 Galileo s Motion Studies gave us Kinematic Equations x v v t 0 f v, v, a v t v v at f 0 1 x v0t at v v a x f 0

61 Quicky Question An automobile enters a freeway on-ramp at 15.0m/s and accelerates uniformly up to 5.0 m/s in a time of 10.0s. a) What is the automobile s average velocity? Which equation? v ave v 0 v f 15 m / s 5 m / s 0 m/ s

62 Quicky Question An automobile enters a freeway on-ramp at 15.0m/s and accelerates uniformly up to 5.0 m/s in a time of 10.0s. b) What is the automobile s average acceleration? Which equation? a v t v f t v i 5 m / s 15 m / s 10s 1 m/ s

63 Quicky Question An automobile enters a freeway on-ramp at 15.0m/s and accelerates uniformly up to 5.0 m/s in a time of 10.0s. c) What is the distance traveled in this amount of time? Which equation? 0 1 x v t at x 1 m 15 / (10 ) 1 (10 ) m s s s s 00m a 1 m/ s (you could also use v ave equation.)

64 Motion Diagrams

65 Skiing through the woods

66 Draw the Motion Diagram An automobile enters a freeway on-ramp at 15.0m/s and accelerates uniformly up to 5.0 m/s in a time of 10.0s.

67 Problem Solving Strategy

68 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? Draw the Motion Graph +x

69 Acceleration: Changing Velocity Knowns a v v i f 5 m / s 8 m / s 0 t? Which equation to use? f Solve for t: v v at i v t f a v 0 8 m/ s 5 m/ s i t 5.6s 5.6s

70 Acceleration: Changing Velocity From t = 0, to t = 5.6s, how far does the car travel before it comes to a stop? Knowns Which equation? 1 a 5 m / s x v t at v v i f 8 m / s 0 t 5.6s m 1 x s ( 5 m / s )(5.6 s) 78.4m s 0 YOU TRY IT! x 78.4m +x

71 Motion Graphs What kind of motion is this?

72 What kind of motion is this?

73 What is the velocity during each segment? v x t v 0 m/ s v 3 400m 400s 1 m / s v 1 400m 00s m / s

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

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

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

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

78 Motion Graphs What is the average velocity between A and B? v x x t x(1 s) x(0 s) (1s 0 s) m 0 1s m s

79 Motion Graphs What is the average velocity between B and D? v x x t x(3 s) x(1 s) (3s 1 s) 6 ( ) m m s 4 m s

80 Rank in order, from largest to smallest, the accelerations a A a C at points A C. A) a A > a B > a C B) a A > a C > a B C) a B > a A > a C D) a C > a A > a B E) a C > a B > a A

81 Rank in order, from largest to smallest, the accelerations a A a C at points A C. A) a A > a B > a C B) a A > a C > a B C) a B > a A > a C D) a C > a A > a B E) a C > a B > a A

82 QuickCheck.1 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. Slide -9

83 QuickCheck.1 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. Slide -30

84 QuickCheck.13 Here is a motion diagram of a car speeding up on a straight road: The sign of the acceleration a x is A. Positive. B. Negative. C. Zero. Slide -69

85 QuickCheck.13 Here is a motion diagram of a car speeding up on a straight road: The sign of the acceleration a x is A. Positive. B. Negative. C. Zero. Speeding up means v x and a x have the same sign. Slide -70

86 QuickCheck.5 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? Slide -44

87 QuickCheck.5 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? Slide -45

88 QuickCheck.7 Here is a position graph of an object: At t = 3.0 s, the object s velocity is A. 40 m/s. B. 0 m/s. C. 10 m/s. D. 10 m/s. E. None of the above. Slide -50

89 QuickCheck.7 Here is a position graph of an object: At t = 3.0 s, the object s velocity is A. 40 m/s. B. 0 m/s. C. 10 m/s. D. 10 m/s. E. None of the above. Slide -51

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

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

92 Galileo s Motion Studies gave us Definitions of averages x v v t 0 f v, v, a v t Kinematic Equations with constant acceleration v v at f 0 1 x v0t at v v a x f 0

93 Newton s Calculus will give us INSTANTEOUS motion

94 Newton s Calculus will give us INSTANTEOUS motion vt ( ) at () dx() t dt dv t dt ( ) d x( t) dt t v( t) a( t) dt 0 x() t t vxdt 0

95 1-D Motion in a nutshell x v v v v f a t 0 Averages:,, dx dv d x Instantaneous: v, a, a dt dt dt Kinematics Eqs: Constant acceleration. v v at f 0 1 x f x0 v0t at v v a x f 0 v f 0 v t Varying acceleration. v v a() t dt f f 0 t x x v() t dt f t 0

96 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!!!!

Finding Position From Velocity 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

97 Finding Position From Velocity 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. 013 Pearson Education, Inc. Slide -55

98 Instantaneous Acceleration 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 -11

99 Slide -114

100 Motion Graphs What kind of motion does this graph represent? What is the NARRATIVE? (story) A-B: Object moves backwards with average speed of m/s, slows down and stops. B-C: C-D: Object moves forward with average speed of m/s, speeding up until it comes back to where it started. Object continues to move forward and increasing speed. Is the acceleration constant or changing during the motion? Find the equation for the displacement. x( t) 4t t

101 Instantaneous Velocity The velocity at any time t is the slope of the x vs t graph at t. v () x t dx dt x( t) 4t t d( 4t t ) v( t) 4 4t dt What is the instantaneous velocity at t=.5s? v(.5 s) 4 m 4 m (.5 s) 6 m s s s What does the velocity vs time graph look like?

102 Velocity Graph x( t) 4t t What does the a-t graph look like?

103 All the Graphs x( t) 4t t v( t) 4 4t a 4 m / s ax 4 m/ s

104 What is the displacement from zero to s? m/s (s) In general

105 Displacement = area under the v-t graph x() t t 0 t 0 v dt x ( axt) dt 1 at x 1 ( axt ) t 1 (base)(height) Area under graph

106 What is the displacement from zero to s? m/s (s) x 1 (base)(height) 1 1 (1 s)(-4 m / s) (1 s)(4 m / s) 0

107 Displacement = area under the v-t graph What is the displacement from zero to s? Zero! 1 x (base)(height) 1 1 (1s)(-4m/s)+ (1s)(4m/s)=0 x( s) 0 x( t) 4t t m/s (s)

108 Displacement = area under the v-t graph What is the displacement from zero to 4s? 1 x (base)(height) 1 1 (1s)(-4m/s)+ (3s)(1m/s)=16m x( t) 4t t x(4 s) 16m m/s (s)

109 1-D Motion in a nutshell x v v v v f a t 0 Averages:,, dx dv d x Instantaneous: v, a, a dt dt dt Kinematics Eqs: Constant acceleration. v v at f 0 1 x f x0 v0t at v v a x f 0 v f 0 v t Varying acceleration. v v a() t dt f f 0 t x x v() t dt f t 0

110 Last Time.. Deriving Graphs from Graphs x( t) 4t t v( t) 4 4t a 4 m / s ax 4 m/ s

111 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.

112 Free Fall Unless told otherwise, ignore air resistance for free fall problems!

113 Galileo Challenged Aristotle Physics In a vacuum, all objects fall with the same acceleration due to gravity: 9.80 m/s, independent of their weight.

114 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/s g decreases with increasing altitude g varies with latitude 9.80 m/s is the average at the Earth s surface We will neglect air resistance g is a SCALAR!!! POSTIVE

115 Free Fall Equations For any object in the absence of air resistance. a g 9.80 m/ s y (taking up as +y) Kinematic Eqs: v v at f 0 1 x v t at 0 0 v v a x f Customize: v v gt f 0 1 y v0t gt v v g y f 0 Note: v 0 can be negative!

116 Falling from Rest v0 0 Estimate : v v gt f a g ~ 10 m / s 0 1 y v0t gt Take down as +y: + v 10 m / s y 5m v 0 m / s y 0m v at 10t 1 y at 5t How FAR is not How FAST! v y! v 30 m / s y 45m v 40 m / s y 80m v 50 m / s y 15m

117 How Far: y(t) ~ t 1 y v t at 0 + How Fast: v(t) ~ t 1 v v at How Fast How Fast is f Changing! g m / s

118 FIRST: Define Reference Frame 9.80 m/ s In this reference frame,what is the sign of a? What is v at t = 3s? Knowns : v0 0, a 9.8 m/ s, t 3s Unknown : v f? a v v at f 0 m (3 s ) s 9.4 m s Negative because it is moving downward, in the negative direction!

119 FIRST: Define Reference Frame How far did the ball fall in those 3 seconds? Knowns v a m s t s v m s Unknown : y? : 0 0, 9.8 /, 3, f 9.4 / 0 1 y v t at 1 m 0 ( 9.8 )(3 ) s s 44.1m The ball fell 44.1m. The displacement is negative because it is moves downward, in the negative direction but how far is a distance a scalar and is positive!

120 Throwing up is Also Free Fall! Symmetry of G Field. Estimate : a g ~ 10 m / s v v gt f 0 1 y v0t gt

121 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?

122 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? Knowns v m s a m s t s Unknown : v f? v v at f 0 : 0 30 /, 9.8 /, 3 +y v f m m (3 ) s s s m 59.4 s How high is the cliff?

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?

123 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? How high is the cliff? Knowns v m s a m s t s Unknown : y? 0 1 y v t at : 0 30 /, 9.8 /, 3 ( 30 m / s)(3 s) ( 9.8 m/ s )(3 s) 134m 1 +y The cliff is 134 m high.

124 Free Fall: Throwing Up What is the speed at the top of the path? ZERO! +y What is the acceleration at the top? a = m/s What is the velocity at the same height on the way down? -30 m/s With what velocity will the rock hit the ground? m/s SAME as if you threw it straight down at 30m/s!

125 Free Fall: Throwing Up Problem How long does it take to hit the ground? First try to guess! How long to the top? How long back to launch point? Final v increases by 30m/s? Knowns v m s a m s t s v m s Unknown : t? : 0 30 /, 9.8 /, 3, f 59.4 / v v at f I guess about 9 seconds! 0 +y t v v 59.4 m / s 30 m / s f 0 a 9.8 m / s t 9.1s

126 QuickCheck.18 A ball is tossed straight up in the air. At its very highest point, the ball s instantaneous acceleration a y is A. Positive. B. Negative. C. Zero. Slide -96

127 QuickCheck.18 A ball is tossed straight up in the air. At its very highest point, the ball s instantaneous acceleration a y is A. Positive. B. Negative. C. Zero. Slide -97

128 Stopping Distance Traveling at 70 miles per hour, what is your breaking distance? v v a x f 0 x 0 v a If v doubles, d quadruples!!! Stopping Distance goes as the SQUARE of the speed!

129 Brake Question You are driving a car going 80 km/hr (50mph) when a head-on collision happens 5 meters ahead of you. If you can brake at 6 m/s, will you hit the crash or stop before it? Knowns : v 80 km / hr m/ s, v 0, a 6 m/ s Unknown : x? 0 f 0 v v a x ( 6 m/ s ) x 0 v a ( m/ s) x 40.3m 5m f Stopping Distance goes as the SQUARE of the speed! CRASH!

130 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.00 m/s 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?

131 HW: Rock Drop A rock is dropped from rest into a well. The sound of the splash is heard 3.0 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.

132 Motion on an Inclined Plane 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. 013 Pearson Education, Inc. Slide -10

133 Sense of Speed 1 m/ s.5 mi / hr 1 m / s 3.6 km / hr.4 mi / hr 10 m / s 36 km / hr.4 mi / hr 0 m / s 7 km / hr 44.8 mi / hr 30 m / s 108 km / hr 67. mi / hr 1 km / hr 0.6 mi / hr 013 Pearson Education, Inc.

Physics 40 1-D Kinematics

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.