Physics 115/242 The Kepler Problem
|
|
- Erik White
- 5 years ago
- Views:
Transcription
1 Physics 115/242 The Kepler Problem Peter Young (Dated: April 21, 23) I. INTRODUCTION We consider motion of a planet around the sun, the Kepler problem, see e.g. Garcia, Numerical Methods for Physics, Sec by Let the planet have mass m and velocity v, and the sun have mass M. The total energy is given E = 1 2 mv2 m, (1) r where r is the distance from the sun to the planet. The energy is conserved, as is the angular momentum: L = m r v. (2) The motion is in a plane so the only non-zero component is The force on the planet is given by L z L = m(xv y yv x ). (3) The negative sign indicates that this is an attractive (inwards) force. F( r) = m r 2 ˆr. (4) II. CIRCULAR MOTION For the special case of circular motion the sum of the gravitational force in Eq. (4) and the centripetal force mv 2 circ /r circ is zero, i.e. so mv 2 circ = m, (5) r circ v circ = r 2 circ r circ. (6)
2 2 It is easy to check that the potential energy is minus twice the kinetic energy so the total energy is E circ = 1 m. (7) 2 r circ The angular momentum, L circ, and period of the circular orbit, T circ, are given by L circ = mv circ r = m r circ (8) T circ = 2π r circ v circ = 2π r 3/2 circ. (9) III. THE GENERAL CASE: ELLIPTICAL MOTION In general the motion is not circular but an ellipse with the sun at one focus, see the figure below. y 2a 2b r θ x q If a and b are the semi-major and semi-minor axes (b a) then the eccentricity, ǫ, is defined to be ǫ = 1 b2 a2. (10) We now quote without proof that the formula for the ellipse is r(θ) = a(1 ǫ2 ) 1 ǫcosθ. (11) Hence, the perihelion (distance at closest approach, θ = π) is given by q = (1 ǫ)a, (12) while the aphelion (distance when furthest away, θ = 0) is given by = (1+ǫ)a. (13)
3 3 We also quote without proof that the eccentricity, angular momentum and energy are related by ǫ = 1+ 2EL2 G 2 M 2 m3. (14) From Eqs. (7) and (8) we see that Eq. (14) correctly gives ǫ = 0 for a circular orbit. One can also show that the period is related to the semi-major axis a by [ T = 2π a3/2 = 2π ( ) ] 3/2, (15) 1+ǫ which is known as Kepler s third law. Furthermore, the energy can also be expressed in terms of a by E = 1 m. (16) 2 a Comparing the general expressions for the energy and period in Eqs. (16) and (15) with those for a circular orbit in Eqs. (7) and (9), we see that they are the same except that r circ is replaced by a in the general case. IV. RELATION TO SPEED AT APHELION In the numerical problems in the homework, we start the planet at the aphelion with speed v i, in a direction perpendicular to that of the star, see the figure below. y v i x q Hence we have L = mv i, E = 1 2 mv2 i m. (17a) (17b)
4 4 We would like to express the eccentricity in terms of v i. To do this, we define v circ to be the speed of a circular orbit of radius, i.e. from Eq. (6), so E in Eq. (17b) can be written as v circ =, (18) E = 1 2 mv2 i mv 2 circ. (19) From Eqs. (17a), (18) and (19), the expression for the eccentricity in Eq. (14) can be written as ǫ 2 = 1+ 2EL2 G 2 M 2 m 3, = 1+ ( v 2 i 2v 2 circ) v 2 i = v 4 circ Hence the eccentricity can be written in the following simple form ( ) 2 1 v2 i vcirc 2. (20) ǫ = 1 v2 i vcirc 2, (21) where we have assumed v i v circ (so that the planet starts at the aphelion rather than the perihelion). Similarly, from Eqs. (11) and (13) we can write and, from Eqs. (9) and (15), r(θ) = 1 ǫ 1 ǫcosθ, (22) T T circ = where T circ is the period of a circular orbit of radius, i.e. from Eq. (9) 1, (23) (1+ǫ) 3/2 T circ = 2π 3/2. (24) Remember that is the aphelion, see the figure.
5 5 V. NUMERICS that In the numerical calculations we set m = M = G = = 1. From the above results it follows v circ = 1, T circ = 2π, ǫ = 1 v 2 i, T = 2π (2 v 2 i )3/2 = r = 1 ǫ 1 ǫcosθ. 2π (1+ǫ) 3/2, (25a) (25b) (25c) (25d) (25e) Orbits of different eccentricity are produced by different values of v i in the region 0 < v i 1. For v i = 1 the orbit is circular (ǫ = 0). If we were to set set v i = 0 (ǫ = 1) the planet would fall into the sun and crash. For ǫ 1, the orbit is close to circular and the speed is nearly constant. However, for ǫ 1, the ellipse becomes more and more squashed, and the magnitude and direction of the velocity change rapidly when the planet approaches the sun. This requires a smaller time step to preserve accuracy. In fact, for ǫ 1, it would be better to use adaptive step-size control so the time-step would be automatically reduced when the planet gets close to the sun and increased again when the planets moves away. (However, adaptive step-size control will not be needed for the homework.)
A = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great Pearson Education, Inc.
Q13.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A. 81 2
More informationNotes 10-3: Ellipses
Notes 10-3: Ellipses I. Ellipse- Definition and Vocab An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F 1 and F
More informationPHYS 106 Fall 2151 Homework 3 Due: Thursday, 8 Oct 2015
PHYS 106 Fall 2151 Homework 3 Due: Thursday, 8 Oct 2015 When you do a calculation, show all your steps. Do not just give an answer. You may work with others, but the work you submit should be your own.
More informationChapter 9. Gravitation
Chapter 9 Gravitation 9.1 The Gravitational Force For two particles that have masses m 1 and m 2 and are separated by a distance r, the force has a magnitude given by the same magnitude of force acts on
More informationLecture 22: Gravitational Orbits
Lecture : Gravitational Orbits Astronomers were observing the motion of planets long before Newton s time Some even developed heliocentric models, in which the planets moved around the sun Analysis of
More informationChapter 13. Gravitation
Chapter 13 Gravitation e = c/a A note about eccentricity For a circle c = 0 à e = 0 a Orbit Examples Mercury has the highest eccentricity of any planet (a) e Mercury = 0.21 Halley s comet has an orbit
More informationKEPLER S LAWS OF PLANETARY MOTION
KEPLER S LAWS OF PLANETARY MOTION In the early 1600s, Johannes Kepler culminated his analysis of the extensive data taken by Tycho Brahe and published his three laws of planetary motion, which we know
More informationCentral force motion/kepler problem. 1 Reducing 2-body motion to effective 1-body, that too with 2 d.o.f and 1st order differential equations
Central force motion/kepler problem This short note summarizes our discussion in the lectures of various aspects of the motion under central force, in particular, the Kepler problem of inverse square-law
More informationChapter 13. Gravitation
Chapter 13 Gravitation 13.2 Newton s Law of Gravitation Here m 1 and m 2 are the masses of the particles, r is the distance between them, and G is the gravitational constant. G =6.67 x10 11 Nm 2 /kg 2
More informationL03: Kepler problem & Hamiltonian dynamics
L03: Kepler problem & Hamiltonian dynamics 18.354 Ptolemy circa.85 (Egypt) -165 (Alexandria) Greek geocentric view of the universe Tycho Brahe 1546 (Denmark) - 1601 (Prague) "geo-heliocentric" system last
More information(b) The period T and the angular frequency ω of uniform rotation are related to the cyclic frequency f as. , ω = 2πf =
PHY 302 K. Solutions for problem set #9. Non-textbook problem #1: (a) Rotation frequency of 1 Hz means one revolution per second, or 60 revolutions per minute (RPM). The pre-lp vinyl disks rotated at 78
More informationKepler s Laws of Orbital Motion. Lecture 5 January 30, 2014
Kepler s Laws of Orbital Motion Lecture 5 January 30, 2014 Parallax If distance is measured in parsecs then d = 1 PA Where PA is the parallax angle, in arcsec NOTE: The distance from the Sun to the Earth
More informationF = ma. G mm r 2. S center
In the early 17 th century, Kepler discovered the following three laws of planetary motion: 1. The planets orbit around the sun in an ellipse with the sun at one focus. 2. As the planets orbit around the
More informationVISUAL PHYSICS ONLINE
VISUAL PHYSICS ONLINE EXCEL SIMULATION MOTION OF SATELLITES DOWNLOAD the MS EXCEL program PA50satellite.xlsx and view the worksheet Display as shown in the figure below. One of the most important questions
More informationChapter 13. Universal Gravitation
Chapter 13 Universal Gravitation Planetary Motion A large amount of data had been collected by 1687. There was no clear understanding of the forces related to these motions. Isaac Newton provided the answer.
More informationName Period Date Earth and Space Science. Solar System Review
Name Period Date Earth and Space Science Solar System Review 1. is the spinning a planetary object on its axis. 2. is the backward motion of planets. 3. The is a unit less number between 0 and 1 that describes
More informationPHY321 Homework Set 10
PHY321 Homework Set 10 1. [5 pts] A small block of mass m slides without friction down a wedge-shaped block of mass M and of opening angle α. Thetriangular block itself slides along a horizontal floor,
More informationUniform Circular Motion
Circular Motion Uniform Circular Motion Uniform Circular Motion Traveling with a constant speed in a circular path Even though the speed is constant, the acceleration is non-zero The acceleration responsible
More informationChapter 13: universal gravitation
Chapter 13: universal gravitation Newton s Law of Gravitation Weight Gravitational Potential Energy The Motion of Satellites Kepler s Laws and the Motion of Planets Spherical Mass Distributions Apparent
More informationUniversal Gravitation
Universal Gravitation Newton s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely
More informationv lim a t = d v dt a n = v2 R curvature
PHY 02 K. Solutions for Problem set # 6. Textbook problem 5.27: The acceleration vector a of the particle has two components, the tangential acceleration a t d v dt v lim t 0 t (1) parallel to the velocity
More informationGravitation. chapter 9
chapter 9 Gravitation Circular orbits (Section 9.3) 1, 2, and 3 are simple exercises to supplement the quantitative calculations of Examples 4, 5, and 6 in Section 9.3. 1. Satellite near Earth s surface
More informationKepler s Laws of Orbital Motion. Lecture 5 January 24, 2013
Kepler s Laws of Orbital Motion Lecture 5 January 24, 2013 Team Extra Credit Two teams: Io & Genius Every class (that is not an exam/exam review) will have a question asked to a random member of each team
More informationLecture 15 - Orbit Problems
Lecture 15 - Orbit Problems A Puzzle... The ellipse shown below has one focus at the origin and its major axis lies along the x-axis. The ellipse has a semimajor axis of length a and a semi-minor axis
More informationChapter 8. Orbits. 8.1 Conics
Chapter 8 Orbits 8.1 Conics Conic sections first studied in the abstract by the Greeks are the curves formed by the intersection of a plane with a cone. Ignoring degenerate cases (such as a point, or pairs
More informationEarth Science Unit 6: Astronomy Period: Date: Elliptical Orbits
Earth Science Name: Unit 6: Astronomy Period: Date: Lab # 5 Elliptical Orbits Objective: To compare the shape of the earth s orbit (eccentricity) with the orbits of and with a circle. other planets Focus
More informationOrbit Characteristics
Orbit Characteristics We have shown that the in the two body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the primary located at the focus of the conic
More informationObservational Astronomy - Lecture 4 Orbits, Motions, Kepler s and Newton s Laws
Observational Astronomy - Lecture 4 Orbits, Motions, Kepler s and Newton s Laws Craig Lage New York University - Department of Physics craig.lage@nyu.edu February 24, 2014 1 / 21 Tycho Brahe s Equatorial
More informationGRAVITATION. F = GmM R 2
GRAVITATION Name: Partner: Section: Date: PURPOSE: To explore the gravitational force and Kepler s Laws of Planetary motion. INTRODUCTION: Newton s law of Universal Gravitation tells us that the gravitational
More informationCRASH COURSE PHYSICS EPISODE #1: Universal gravitation
CRASH COURSE PHYSICS EPISODE #1: Universal gravitation No one (including me) really seems to understand physics when Bordak teaches so I decided to translate all the Universal Gravitation stuff to plain
More informationKepler's Laws and Newton's Laws
Kepler's Laws and Newton's Laws Kepler's Laws Johannes Kepler (1571-1630) developed a quantitative description of the motions of the planets in the solar system. The description that he produced is expressed
More informationChapter 5 Centripetal Force and Gravity. Copyright 2010 Pearson Education, Inc.
Chapter 5 Centripetal Force and Gravity v Centripetal Acceleration v Velocity is a Vector v It has Magnitude and Direction v If either changes, the velocity vector changes. Tumble Buggy Demo v Centripetal
More informationSatellite Communications
Satellite Communications Lecture (3) Chapter 2.1 1 Gravitational Force Newton s 2nd Law: r r F = m a Newton s Law Of Universal Gravitation (assuming point masses or spheres): Putting these together: r
More informationA study upon Eris. I. Describing and characterizing the orbit of Eris around the Sun. I. Breda 1
Astronomy & Astrophysics manuscript no. Eris c ESO 2013 March 27, 2013 A study upon Eris I. Describing and characterizing the orbit of Eris around the Sun I. Breda 1 Faculty of Sciences (FCUP), University
More informationResonance In the Solar System
Resonance In the Solar System Steve Bache UNC Wilmington Dept. of Physics and Physical Oceanography Advisor : Dr. Russ Herman Spring 2012 Goal numerically investigate the dynamics of the asteroid belt
More informationToday. Events. Energy. Gravity. Homework Due Next time. Practice Exam posted
Today Energy Gravity Events Homework Due Next time Practice Exam posted Autumn is here! Autumnal equinox occurred at 11:09pm last night night and day very nearly equal today days getting shorter Moon is
More informationPhysics 12. Unit 5 Circular Motion and Gravitation Part 2
Physics 12 Unit 5 Circular Motion and Gravitation Part 2 1. Newton s law of gravitation We have seen in Physics 11 that the force acting on an object due to gravity is given by a well known formula: F
More information5.1. Accelerated Coordinate Systems:
5.1. Accelerated Coordinate Systems: Recall: Uniformly moving reference frames (e.g. those considered at 'rest' or moving with constant velocity in a straight line) are called inertial reference frames.
More informationGravitation. Makes the World Go Round
Gravitation Makes the World Go Round Gravitational Force The Force of gravity is an attractive force felt between all objects that have mass. G=6.67x10-11 N m 2 /kg 2 Example 1: What is the Force of Gravity
More informationCopyright 2009, August E. Evrard.
Unless otherwise noted, the content of this course material is licensed under a Creative Commons BY 3.0 License. http://creativecommons.org/licenses/by/3.0/ Copyright 2009, August E. Evrard. You assume
More informationA SIMULATION OF THE MOTION OF AN EARTH BOUND SATELLITE
DOING PHYSICS WITH MATLAB A SIMULATION OF THE MOTION OF AN EARTH BOUND SATELLITE Download Directory: Matlab mscripts mec_satellite_gui.m The [2D] motion of a satellite around the Earth is computed from
More information1 Summary of Chapter 2
General Astronomy (9:61) Fall 01 Lecture 7 Notes, September 10, 01 1 Summary of Chapter There are a number of items from Chapter that you should be sure to understand. 1.1 Terminology A number of technical
More informationAP Physics C Textbook Problems
AP Physics C Textbook Problems Chapter 13 Pages 412 416 HW-16: 03. A 200-kg object and a 500-kg object are separated by 0.400 m. Find the net gravitational force exerted by these objects on a 50.0-kg object
More informationGravitation & Kepler s Laws
Gravitation & Kepler s Laws What causes YOU to be pulled down to the surface of the earth? THE EARTH.or more specifically the EARTH S MASS. Anything that has MASS has a gravitational pull towards it. F
More informationThe Heliocentric Model of Copernicus
Celestial Mechanics The Heliocentric Model of Copernicus Sun at the center and planets (including Earth) orbiting along circles. inferior planets - planets closer to Sun than Earth - Mercury, Venus superior
More informationLecture 13. Gravity in the Solar System
Lecture 13 Gravity in the Solar System Guiding Questions 1. How was the heliocentric model established? What are monumental steps in the history of the heliocentric model? 2. How do Kepler s three laws
More informationNewton s Gravitational Law
1 Newton s Gravitational Law Gravity exists because bodies have masses. Newton s Gravitational Law states that the force of attraction between two point masses is directly proportional to the product of
More informationAST111 PROBLEM SET 2 SOLUTIONS. RA=02h23m35.65s, DEC=+25d18m42.3s (Epoch J2000).
AST111 PROBLEM SET 2 SOLUTIONS Home work problems 1. Angles on the sky and asteroid motion An asteroid is observed at two different times. The asteroid is located at RA=02h23m35.65s, DEC=+25d18m42.3s (Epoch
More informationCelestial Mechanics Lecture 10
Celestial Mechanics Lecture 10 ˆ This is the first of two topics which I have added to the curriculum for this term. ˆ We have a surprizing amount of firepower at our disposal to analyze some basic problems
More informationASTRO 1050 LAB #3: Planetary Orbits and Kepler s Laws
ASTRO 1050 LAB #3: Planetary Orbits and Kepler s Laws ABSTRACT Johannes Kepler (1571-1630), a German mathematician and astronomer, was a man on a quest to discover order and harmony in the solar system.
More informationSatellite meteorology
GPHS 422 Satellite meteorology GPHS 422 Satellite meteorology Lecture 1 6 July 2012 Course outline 2012 2 Course outline 2012 - continued 10:00 to 12:00 3 Course outline 2012 - continued 4 Some reading
More information18. Kepler as a young man became the assistant to A) Nicolaus Copernicus. B) Ptolemy. C) Tycho Brahe. D) Sir Isaac Newton.
Name: Date: 1. The word planet is derived from a Greek term meaning A) bright nighttime object. B) astrological sign. C) wanderer. D) nontwinkling star. 2. The planets that were known before the telescope
More informationLecture 16. Gravitation
Lecture 16 Gravitation Today s Topics: The Gravitational Force Satellites in Circular Orbits Apparent Weightlessness lliptical Orbits and angular momentum Kepler s Laws of Orbital Motion Gravitational
More informationCopyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Chapter 13. Newton s Theory of Gravity The beautiful rings of Saturn consist of countless centimeter-sized ice crystals, all orbiting the planet under the influence of gravity. Chapter Goal: To use Newton
More informationToday. Laws of Motion. Conservation Laws. Gravity. tides
Today Laws of Motion Conservation Laws Gravity tides Newton s Laws of Motion Our goals for learning: Newton s three laws of motion Universal Gravity How did Newton change our view of the universe? He realized
More informationQuestion 1. GRAVITATION UNIT H.W. ANS KEY
Question 1. GRAVITATION UNIT H.W. ANS KEY Question 2. Question 3. Question 4. Two stars, each of mass M, form a binary system. The stars orbit about a point a distance R from the center of each star, as
More informationGravity: Motivation An initial theory describing the nature of the gravitational force by Newton is a product of the resolution of the
Gravity: Motivation An initial theory describing the nature of the gravitational force by Newton is a product of the resolution of the Geocentric-Heliocentric debate (Brahe s data and Kepler s analysis)
More informationAP Physics QUIZ Gravitation
AP Physics QUIZ Gravitation Name: 1. If F1 is the magnitude of the force exerted by the Earth on a satellite in orbit about the Earth and F2 is the magnitude of the force exerted by the satellite on the
More informationGravitation. Kepler s Law. BSc I SEM II (UNIT I)
Gravitation Kepler s Law BSc I SEM II (UNIT I) P a g e 2 Contents 1) Newton s Law of Gravitation 3 Vector representation of Newton s Law of Gravitation 3 Characteristics of Newton s Law of Gravitation
More informationr p L = = So Jupiter has the greater angular momentum.
USAAAO 2015 Second Round Solutions Problem 1 T 10000 K, m 5, d 150 pc m M 5 log(d/10) M m 5 log(d/10) 5 5 log(15) 0.88 We compare this with the absolute magnitude of the sun, 4.83. solar 100 (4.83 0.88)/5
More informationPhysics 103 Homework 5 Summer 2015
Physics 103 Homework 5 Summer 2015 Instructor: Keith Fratus TAs: Michael Rosenthal and Milind Shyani Due Date: Friday, July 31 st, IN LECTURE 1. A particle moving in three dimensions is constrained to
More informationExact Values of Dynamical Quantities in Planetary Motion
International Journal of Mathematical Education. ISSN 0973-6948 Volume 6, Number 1 (2016), pp. 1-9 Research India Publications http://www.ripublication.com Exact Values of Dynamical Quantities in Planetary
More informationAST111, Lecture 1b. Measurements of bodies in the solar system (overview continued) Orbital elements
AST111, Lecture 1b Measurements of bodies in the solar system (overview continued) Orbital elements Planetary properties (continued): Measuring Mass The orbital period of a moon about a planet depends
More informationLecture 9(+10) Physics 106 Spring 2006
3/22/2006 Andrei Sirenko, NJIT 3 3/22/2006 Andrei Sirenko, NJIT 4 Lecture 9(+10) Physics 106 Spring 2006 Gravitation HW&R http://web.njit.edu/~sirenko/ Gravitation On Earth: the Earth gravitation dominates
More informationChapter 13. Gravitation. PowerPoint Lectures for University Physics, 14th Edition Hugh D. Young and Roger A. Freedman Lectures by Jason Harlow
Chapter 13 Gravitation PowerPoint Lectures for University Physics, 14th Edition Hugh D. Young and Roger A. Freedman Lectures by Jason Harlow Next one week Today: Ch 13 Wed: Review of Ch 8-11, focusing
More informationPhys 7221, Fall 2006: Midterm exam
Phys 7221, Fall 2006: Midterm exam October 20, 2006 Problem 1 (40 pts) Consider a spherical pendulum, a mass m attached to a rod of length l, as a constrained system with r = l, as shown in the figure.
More informationJohannes Kepler ( ) German Mathematician and Astronomer Passionately convinced of the rightness of the Copernican view. Set out to prove it!
Johannes Kepler (1571-1630) German Mathematician and Astronomer Passionately convinced of the rightness of the Copernican view. Set out to prove it! Kepler s Life Work Kepler sought a unifying principle
More informationTP 3:Runge-Kutta Methods-Solar System-The Method of Least Squares
TP :Runge-Kutta Methods-Solar System-The Method of Least Squares December 8, 2009 1 Runge-Kutta Method The problem is still trying to solve the first order differential equation dy = f(y, x). (1) dx In
More informationAP Physics 1 Chapter 7 Circular Motion and Gravitation
AP Physics 1 Chapter 7 Circular Motion and Gravitation Chapter 7: Circular Motion and Angular Measure Gravitation Angular Speed and Velocity Uniform Circular Motion and Centripetal Acceleration Angular
More informationGravitation and Newton s Synthesis
Lecture 10 Chapter 6 Physics I 0.4.014 Gravitation and Newton s Synthesis Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Lecture Capture: http://echo360.uml.edu/danylov013/physics1spring.html
More informationTopic 6 The Killers LEARNING OBJECTIVES. Topic 6. Circular Motion and Gravitation
Topic 6 Circular Motion and Gravitation LEARNING OBJECTIVES Topic 6 The Killers 1. Centripetal Force 2. Newton s Law of Gravitation 3. Gravitational Field Strength ROOKIE MISTAKE! Always remember. the
More informationThe Two -Body Central Force Problem
The Two -Body Central Force Problem Physics W3003 March 6, 2015 1 The setup 1.1 Formulation of problem The two-body central potential problem is defined by the (conserved) total energy E = 1 2 m 1Ṙ2 1
More informationHow does the solar system, the galaxy, and the universe fit into our understanding of the cosmos?
Remember to check the links for videos! How does the solar system, the galaxy, and the universe fit into our understanding of the cosmos? Universe ~ 13.7 bya First Stars ~ 13.3 bya First Galaxies ~ 12.7
More informationHOMEWORK AND EXAMS. Homework Set 13 due Wednesday November 29. Exam 3 Monday December 4. Homework Set 14 due Friday December 8
HOMEWORK AND EXAMS Homework Set 13 due Wednesday November 29 Exam 3 Monday December 4 Homework Set 14 due Friday December 8 Final Exam Tuesday December 12 1 Section 8.6. The Kepler orbits Read Section
More informationSkills Practice Skills Practice for Lesson 12.1
Skills Practice Skills Practice for Lesson.1 Name Date Try to Stay Focused Ellipses Centered at the Origin Vocabulary Match each definition to its corresponding term. 1. an equation of the form a. ellipse
More information9.2 GRAVITATIONAL FIELD, POTENTIAL, AND ENERGY 9.4 ORBITAL MOTION HW/Study Packet
9.2 GRAVITATIONAL FIELD, POTENTIAL, AND ENERGY 9.4 ORBITAL MOTION HW/Study Packet Required: READ Hamper pp 186-191 HL Supplemental: Cutnell and Johnson, pp 144-150, 169-170 Tsokos, pp 143-152 Tsokos Questions
More informationGravitation and Newton s Synthesis
Lecture 10 Chapter 6 Physics I 0.4.014 Gravitation and Newton s Synthesis Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Lecture Capture: http://echo360.uml.edu/danylov013/physics1spring.html
More informationVISUAL PHYSICS ONLINE
VISUAL PHYSICS ONLINE PRACTICAL ACTIVITY HOW DO THE PANETS MOVE? One of the most important questions historically in Physics was how the planets move. Many historians consider the field of Physics to date
More informationPart I Multiple Choice (4 points. ea.)
ach xam usually consists of 10 ultiple choice questions which are conceptual in nature. They are often based upon the assigned thought questions from the homework. There are also 4 problems in each exam,
More informationUse conserved quantities to reduce number of variables and the equation of motion (EOM)
Physics 106a, Caltech 5 October, 018 Lecture 8: Central Forces Bound States Today we discuss the Kepler problem of the orbital motion of planets and other objects in the gravitational field of the sun.
More informationStellar Dynamics and Structure of Galaxies
Stellar Dynamics and Structure of Galaxies in a given potential Vasily Belokurov vasily@ast.cam.ac.uk Institute of Astronomy Lent Term 2016 1 / 59 1 Collisions Model requirements 2 in spherical 3 4 Orbital
More informationCH 8. Universal Gravitation Planetary and Satellite Motion
CH 8 Universal Gravitation Planetary and Satellite Motion Sir Isaac Newton UNIVERSAL GRAVITATION Newton: Universal Gravitation Newton concluded that earthly objects and heavenly objects obey the same physical
More informationIntroduction to Astronomy
Introduction to Astronomy AST0111-3 (Astronomía) Semester 2014B Prof. Thomas H. Puzia Newton s Laws Big Ball Fail Universal Law of Gravitation Every mass attracts every other mass through a force called
More informationGravitation and the Waltz of the Planets
Gravitation and the Waltz of the Planets Chapter Four Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets
More informationGravitation and the Waltz of the Planets. Chapter Four
Gravitation and the Waltz of the Planets Chapter Four Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets
More informationBasics of Kepler and Newton. Orbits of the planets, moons,
Basics of Kepler and Newton Orbits of the planets, moons, Kepler s Laws, as derived by Newton. Kepler s Laws Universal Law of Gravity Three Laws of Motion Deriving Kepler s Laws Recall: The Copernican
More informationName. Satellite Motion Lab
Name Satellite Motion Lab Purpose To experiment with satellite motion using an interactive simulation in order to gain an understanding of Kepler s Laws of Planetary Motion and Newton s Law of Universal
More informationPHYSICS 12 NAME: Gravitation
NAME: Gravitation 1. The gravitational force of attraction between the Sun and an asteroid travelling in an orbit of radius 4.14x10 11 m is 4.62 x 10 17 N. What is the mass of the asteroid? 2. A certain
More informationConic Sections in Polar Coordinates
Conic Sections in Polar Coordinates MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction We have develop the familiar formulas for the parabola, ellipse, and hyperbola
More informationJanuary 21, 2018 Math 9. Geometry. The method of coordinates (continued). Ellipse. Hyperbola. Parabola.
January 21, 2018 Math 9 Ellipse Geometry The method of coordinates (continued) Ellipse Hyperbola Parabola Definition An ellipse is a locus of points, such that the sum of the distances from point on the
More informationThe Restricted 3-Body Problem
The Restricted 3-Body Problem John Bremseth and John Grasel 12/10/2010 Abstract Though the 3-body problem is difficult to solve, it can be modeled if one mass is so small that its effect on the other two
More informationLecture 1a: Satellite Orbits
Lecture 1a: Satellite Orbits Meteorological Satellite Orbits LEO view GEO view Two main orbits of Met Satellites: 1) Geostationary Orbit (GEO) 1) Low Earth Orbit (LEO) or polar orbits Orbits of meteorological
More informationAP Physics Multiple Choice Practice Gravitation
AP Physics Multiple Choice Practice Gravitation 1. Each of five satellites makes a circular orbit about an object that is much more massive than any of the satellites. The mass and orbital radius of each
More informationGravitation and the Motion of the Planets
Gravitation and the Motion of the Planets 1 Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets go around
More informationNotes on Planetary Motion
(1) Te motion is planar Notes on Planetary Motion Use 3-dimensional coordinates wit te sun at te origin. Since F = ma and te gravitational pull is in towards te sun, te acceleration A is parallel to te
More informationNewton s Laws and the Nature of Matter
Newton s Laws and the Nature of Matter The Nature of Matter Democritus (c. 470-380 BCE) posited that matter was composed of atoms Atoms: particles that can not be further subdivided 4 kinds of atoms: earth,
More informationAstronomy 111 Exam Review Problems (Real exam will be Tuesday Oct 25, 2016)
Astronomy 111 Exam Review Problems (Real exam will be Tuesday Oct 25, 2016) Actual Exam rules: you may consult only one page of formulas and constants and a calculator while taking this test. You may not
More informationTangent and Normal Vectors
Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in
More informationChapter 14 Satellite Motion
1 Academic Physics Mechanics Chapter 14 Satellite Motion The Mechanical Universe Kepler's Three Laws (Episode 21) The Kepler Problem (Episode 22) Energy and Eccentricity (Episode 23) Navigating in Space
More information