Physics Part 1 MECHANICS Draft (part C incomplete) Topic II. Motion in one Dimension (Kinematics) W. Pezzaglia Updated: 01Aug3 II. Motion in 1D A. Principle of Inertia B. Uniform Motion C. Acceleration A. Principle of Inertia 3 1. Aristotle s Physics 4 1. Aristotle s Physics. Inertia 3. Motion is Relative Natural state is rest continuation of motion depends on continued action of a force [e.g. the prime mover (god) must continue to push the moon] Antiperistasis: To explain how a ball rolls when no-one is pushing it: as ball rolls, it leaves empty space behind it. Nature abhors a vacuum, so air rushes in to fill the space vacated by the ball, hence continues to push the ball along, but it will eventually stop. Aristotle: 384 BC-3 BC. Inertia 5 b. Natural Motion is Straight 6 (a) Impetus Theory: Giambattista Benedetti 1530-1590 Jean Buridan proposes motion continues because the mover has imparted some impetus to the object (which dissipates) Natural motion is straight, not circular When projectile released from a sling, it goes straight. Suggests planets have circular impetus, continuing their orbits forever as there is no resistance in the heavens. His work influences Galileo If string breaks, ball goes straight Jean Buridan 1300-1358 1
c. Galileo s Law of Inertia Bodies at rest tend to stay at rest Bodies in uniform motion will tend to stay in uniform motion Unless acted upon by an outside force natural state IS motion Galileo Galilei 1564-164 Demo: Play 11:35-14:5 of MU #4: http://video.google.com/videoplay?docid=776979968684669875&hl=en&emb=1 7 3. Motion is Relative (a) Absolute Motion Aristotle (350 BC) thought that the earth is at rest, the stars, sun and planets revolve around us Aristarchus (50 BC) proposed that since the sun was bigger, that the earth goes around the sun. Followers of Aristotle (e.g. Hipparchus 150 BC) argued that the earth cannot be moving, else falling bodies would fall sideways. Hence, the earth must be at absolute rest. Geocentric Theory of the universe 8 3b. The Earth Moves Copernicus (1543 AD) shows that the complex motion of the planets can be more simply explained if the sun is at rest, and the earth revolves around it ( heliocentric theory ). However, he doesn t really address the issue of why falling bodies don t fall sideways if the earth is moving. Geocentric Theory of the universe 9 3c. Galileo: no absolute motion 10 Galileo: (163) Motion is relative. A ball dropped from the crow s nest will hit at the base of the ship s mast, even if the ship is moving. Hence, there is no absolute measure of motion (or rest). Demo: Play 11:35-0:3 of MU #4: http://video.google.com/videoplay?docid=776979968684669875&hl=en&emb=1 B. Motion 11 1. Motion in Time 1 1. Motion in Time. Average Velocity 3. Instantaneous Velocity What IS time? Time is what happens when nothing else does. Why does time only flow forward? Are all measurements of time indirect? (i.e. involve motion)
a. Descarte s Graph 13 b. Motion in Time 14 Cogito ergo sum (I think, therefore I am) Galileo is the first (I think) to represent motion by a graph of position with respect to time 1637 Cartesian Coordinates Geometry could be represented by algebraic equations Hence path of motion (e.g. orbit of moon) could be described by an equation and plotted on a graph. Rene Descartes 1596-1650 An Event is a point in spacetime (x,t), saying the object is at position x at time t Worldline of an object is a sequence of events c. Velocity 15. Average Velocity 16 Slope of worldline is velocity x v SI Units: meters/second (a) Average Speed Avg Speed Total Distance Total Time Equation of uniform motion (initial position x 0 at time zero) x( t) 0 x vt Total distance 10 m Total time: 15 seconds Avg Speed=10/15= 8 m/s What is the average speed here? b. Displacement 17 c. (Average) Velocity 18 Velocity is speed with a direction Displacement is the change in position Avg Velocity Total Displacement Total Time 0 to 5 s x=+60 m Time Interval 0 to 5 s Average Velocity v= 1 m/s 0 to 10 s x=+60 m 0 to 10 s v= 6 m/s 10 to 15 s x=-60 m Note minus displacement means movement backwards! 10 to 15 s v= -1 m/s 0 to 15 s x=+0 m 0 to 15 s v= 0 m/s The average velocity is zero because you end up where you started (but the average speed is NOT zero) 3
3. Instantaneous Velocity 19 b. Secant Line 0 (a) Tangent Line Generally velocity changes with time. Its impossible to measure instantaneous velocity. Finite time must pass. Instantaneous velocity at a point is the slope of the tangent line Slope of secant line approximates instantaneous velocity if is small. x x( t ) x( t) v(t) Lim Lim 0 0 c. The Derivative Instantaneous velocity is actually the derivative. Finding exact equations for tangent lines was the beginning of calculus 1 C. Acceleration 1. Falling Motion. Uniform Acceleration Example: Parabola x(t) kt x(t ) k x( t ) x( t) v(t) kt kt kt k 3. Kinematic Equations 3. Acceleration 3 1. Falling Motion 4 Definition Acceleration is the rate of change of velocity (i.e. change in velocity with respect to a change in time) Δv a Δt v Change in velocity Change in time In first 30 seconds the velocity has gone from zero to 10 m/s. What is the acceleration? What is the acceleration from 30 to 45 sec? m Δv 10 s a 0.33 Δt 30 s Zero! (no change in velocity) m s a) Aristotle s Theory b) Galileo s Law of Falling Bodies c) Einstein s theory of gravity 4
Law of Falling: Introduction 5 (a) Aristotle and Falling Motion 6 the rate of falling is proportional to the weight and inversely proportional to the density of the medium the peripatetics (followers of Aristotle) postulated that the speed was proportional to the distance fallen. x v x Aristotle: 384 BC-3 BC Aristotle: Heavier balls will fall faster Galileo: They fall at the same rate! Galileo shows that this must be wrong, for an object starting at zero speed would never acquire any speed! (b) Galileo s Experiment at Pisa 1590 Galileo s Principle: all bodies fall at the same rate, regardless of mass 1907 Strong EEP (Einstein Equivalence Principle) same result, but Einstein argued from a different way. 7 (c) The Einstein Equivalence Principle Reference at rest with Gravity is indistinguishable to a reference frame which is accelerating upward in gravity free environment. 8 Einstein proposed that falling bodies in gravity are equivalent to being in an accelerated frame (e.g. in an accelerating elevator) The apple accelerating downward due to gravity looks the same as an apple at rest in space, with the floor accelerating upward towards it.. Uniform Acceleration 9 a. Inclined Plane Experiments 30 a) Galileo s Inclined Plane b) The law of squares c) Acceleration and velocity Falling motion is too fast to measure Galileo shows rolling balls down inclined plane is same type of motion, but easier to measure because slower Discovers falling motion has constant acceleration 5
b Galileo s Law of Squares 31 c. Results 3 Total distance d traveled is proportional to square of time Equation: d 1 at Where a is the acceleration For gravity, a=g= 9.8 m/s 3. Kinematic Equations 33 3. Kinematic Equations 34 a) Velocity and Instantaneous Acceleration For constant acceleration (falling motion, or ball rolling down inclined plane): b) Distance and (uniform) acceleration Velocity increases linearly with time: v at c) The third kinematic equation Distance increases with square of time: Hence distance is related to velocity: d 1 at v ad References Descartes, R. La Géométrie. Livre Premier: Des problèmes qu'on peut construire sans y employer que des cercles et des lignes droites (Book one: Problems whose construction requires only circles and straight lines). (French) Galileo, Two New Sciences (1638). Chapter Third Day discusses motion and acceleration. Origin of terms abscissa and ordinate, see http://www.pballew.net/arithme1.html#abscissa SAT Physics: http://www.sparknotes.com/testprep/books/sat/physics/chapter5.rhtml Good Java Demo: http://www.jimloy.com/cindy/galilean.htm Video References Inclined Plane short video: http://catalogue.museogalileo.it/multimedia/inclinedplane.html Video: Mechanical Universe #: Law of Falling bodies: http://video.google.com/videoplay?docid=3641917188010584794# Video: Mechanical Universe #4: Inertia: http://video.google.com/videoplay?docid=776979968684669875&hl=en&emb=1 Galileo interview: http://www.youtube.com/watch?v=ygtoxncow1s Galileo: http://www.youtube.com/watch?v=mnmzlcyij-8 Video: Losey's 1975 adaptation of Brecht's play about Galileo Galilei Part1, Invents (steals) the telescope Part, sells telescope to Venice, mountains on moon, moons of jupiter Part3, The young duke refuses to look through telescope, first problem with church, Clavius says he's right Part4, Meets cardinals Barberini and Bellarmine (that Copernicus is wrong) Part5, Student priest has crises Part6, needle floats, aristotle is wrong, sunspots, pope is dying Part7, observe sunspots, play within play, more trouble with church Part8, before inquisition, pope Barberini, recantation Part9, under house arrest, smuggling out book Two New Sciences Part10, ditribe on ethics of science 6