Geodesics as gravity
|
|
- Lisa Russell
- 5 years ago
- Views:
Transcription
1 Geodesics as ravity February 8, 05 It is not obvious that curvature can account for ravity. The orbitin path of a planet, for example, does not immediately seem to be the shortest path between points. Even more immediate is the way the motion of projectiles differs dependin on initial conditions. Consider the tihtly parabolic path followed by a stone thrown hih into the air, contrasted with the nearly straiht-line path of an arrow. How can these two paths, which may pass throuh nearly identical reions of space, be rearded as tracin eodesics in the same curved eometry? The answer lies in the importance of time in ivin a 4-dimensional picture. The curvature of a curve Suppose we have a curve lyin in the xy-plane, iven by y y x). At any point P x 0, y x 0 )) of the curve we place a circle. The best fit circle is the one which is tanent at P, and matches the curve as nearly as possible. To find this circle, choose coordinates with oriin at P and the x axis tanent at P. Then dy y x 0 ) y 0) 0, and the slope of the curve vanishes, 0. Now expand y x)near P. By our dx 0 choice of coordinates, the first two terms in the series vanish, y curve x) y x 0 ) + dy dx x x 0 ) + d y! dx x x 0 ) + d y! dx x + Suppose, for concreteness, that at P the curve has d y dx > 0. Now define a circle of radius R, tanent to the curve at P. The center of the circle will be a distance R up the y axis and consist of point satisfyin Solvin this for y circle x), we have x + y R) R y circle R ± R x To et points near the oriin, we require the lower sin, and expand in a power series, y circle R R x R R x R R R x ) R + x R +
2 The best fit circle is the circle determined by matchin these curves: y curve x) d y! dx x + y circle x R + In eneral, the hiher derivatives will not match, but we can match the ond derivative by choosin the radius of the circle: x R d y! dx x R d y dx We define the curvature at any point of the curve to be R where R is the radius of the best fit circle. The curvature of projectile motion Consider the two motions: a stone is thrown upward at a steep anle θ, landin at a distance d, while another is thrown at hih speed at a taret in the same final location. Each stone follows a parabolic motion: x v 0x t y v 0y t t so that ) x y x) v 0y ) x v 0x v 0x The distance traveled, x d, occurs when y 0: ) d 0 v 0y ) d v 0x v 0x d v 0xv 0y If the projectile is launched at an anle iven by tan θ v0y v 0x and The maximum heiht is reached at when v y 0, v 0y v 0x tan θ then d v 0x tan θ d v 0x tan θ v 0y v 0x tan θ d tan θ 0 v 0y t t v 0y
3 and at this time we have h v 0y d 4 tan θ We find the curvature at the top of the trajectory, P x top, h) d, d 4 tan θ). At any point, the curve may be described by x x 0 + v 0x t y y 0 + v oy t t At the top, x 0, y 0 ) d, d 4 tan θ) and the initial velocity is v 0x, v 0y ) v ox, 0). Therefore, x d + v 0xt and the curve y x) is t y d 4 tan θ t v 0x x d ) tan θ d x d ) y d 4 tan θ tan θ d d 4 tan θ tan θ d x d x d ) The best fit circle will match the ond derivative: R tan θ d and we see that the curvature is lare for steep anles and small for shallow anles. There is a dramatic difference in the curvature for different initial conditions. Suppose the total time of fliht, t v 0y ) d tan θ is ond for one stone and 0 onds for another over a distance d 5 meters. Then for the first 50 0 tan θ tan θ 5 the curvature at the top is tan θ R d
4 while for the ond, and the curvature is tan θ tan θ 0 40 R The curvatures differ by a factor of 00 for ordinary trajectories. This makes is impossible to model the motions of the stones by lettin them follow eodesics in a curved 3-dimensional eometry. To build a eometric model, we need the curvature to be very nearly the same for the two stones, so they both move alon comparably shaped paths. 3 The curvature in spacetime Now place the motion in spacetime. The non-relativistic) projectile follows a curve parameterized by time: ct, x, y) ct, v 0x t, ) t The motion lies in a plane rotated in the xt plane at an anle with tan ϕ v0x c proper distance proper time!) alon this direction. Then cτ c t v0x t ct tan ϕ τ t. Let τ c t x be Then we may parameterize the curve in this plane by cτ, y) cτ, τ ) ) λ, λ c where we parameterize with λ cτ. The best-fit circle to this curve ives c τ R c τ c R c and this is independent of initial conditions. This means that the hih arc of a slow stone, and the flat trajectory of a fast stone have the same curvature in spacetime! Both trajectories could lie on a spherically curved surface where the sphere has a hue radius, c, so it is only ently curved. In sharp contrast to the Euclidean case, where a factor of 0 difference in time of fliht requires a factor of 00 difference in radius of curvature, even a difference of a factor of 0 4 in time of fliht makes completely neliible difference to the curvature required in spacetime. 4
5 Exerise: Find the difference in curvatures m R between initial velocities of 3 Answer: If we don t nelect terms of order v c, then t and τ are related by cτ c t v0x t so ct τ t v 0x c v 0x c τ t v 0x c Now, when we parameterize the curve by cτ, we have ) cτ, y) cτ, t cτ, cτ c v 0x c Therefore, the curvature is iven exactly by c τ R c τ c v 0x c R c v 0x c Now consider stones thrown with velocities of 3 m v 0x c and 3 04 m m and In the first case, so the curvature is R c v 0x c ) For the ond stone, v 0x c
6 and the curvature is R c v 0x c ) The difference between these neliible. 6
v( t) g 2 v 0 sin θ ( ) ( ) g t ( ) = 0
PROJECTILE MOTION Velocity We seek to explore the velocity of the projectile, includin its final value as it hits the round, or a taret above the round. The anle made by the velocity vector with the local
More information2.2 Differentiation and Integration of Vector-Valued Functions
.. DIFFERENTIATION AND INTEGRATION OF VECTOR-VALUED FUNCTIONS133. Differentiation and Interation of Vector-Valued Functions Simply put, we differentiate and interate vector functions by differentiatin
More informationPROJECTILE MOTION. ( ) g y 0. Equations ( ) General time of flight (TOF) General range. Angle for maximum range ("optimum angle")
PROJECTILE MOTION Equations General time of fliht (TOF) T sin θ y 0 sin( θ) General rane R cos( θ) T R cos θ sin( θ) sin( θ) y 0 Anle for maximum rane ("optimum anle") θ opt atan y 0 atan v f atan v f
More informationREVIEW: Going from ONE to TWO Dimensions with Kinematics. Review of one dimension, constant acceleration kinematics. v x (t) = v x0 + a x t
Lecture 5: Projectile motion, uniform circular motion 1 REVIEW: Goin from ONE to TWO Dimensions with Kinematics In Lecture 2, we studied the motion of a particle in just one dimension. The concepts of
More information2.5 Velocity and Acceleration
82 CHAPTER 2. VECTOR FUNCTIONS 2.5 Velocity and Acceleration In this section, we study the motion of an object alon a space curve. In other words, as the object moves with time, its trajectory follows
More informationPhysics 11 Fall 2012 Practice Problems 2 - Solutions
Physics 11 Fall 01 Practice Problems - s 1. True or false (inore any effects due to air resistance): (a) When a projectile is fired horizontally, it takes the same amount of time to reach the round as
More informationKINEMATICS PREVIOUS EAMCET BITS ENGINEERING PAPER
KINEMATICS PREVIOUS EAMCET BITS ENGINEERING PAPER. A body is projected vertically upwards at time t = 0 and is seen at a heiht at time t and t seconds durin its fliht. The maximum heiht attained is [ =
More information(a) 1m s -2 (b) 2 m s -2 (c) zero (d) -1 m s -2
11 th Physics - Unit 2 Kinematics Solutions for the Textbook Problems One Marks 1. Which one of the followin Cartesian coordinate system is not followed in physics? 5. If a particle has neative velocity
More informationXI PHYSICS M. AFFAN KHAN LECTURER PHYSICS, AKHSS, K. https://promotephysics.wordpress.com
XI PHYSICS M. AFFAN KHAN LECTURER PHYSICS, AKHSS, K affan_414@live.com https://promotephysics.wordpress.com [MOTION IN TWO DIMENSIONS] CHAPTER NO. 4 In this chapter we are oin to discuss motion in projectile
More informationVector Valued Functions
SUGGESTED REFERENCE MATERIAL: Vector Valued Functions As you work throuh the problems listed below, you should reference Chapters. &. of the recommended textbook (or the equivalent chapter in your alternative
More informationGround Rules. PC1221 Fundamentals of Physics I. Position and Displacement. Average Velocity. Lectures 7 and 8 Motion in Two Dimensions
PC11 Fundamentals of Physics I Lectures 7 and 8 Motion in Two Dimensions Dr Tay Sen Chuan 1 Ground Rules Switch off your handphone and paer Switch off your laptop computer and keep it No talkin while lecture
More information7.2 Maximization of the Range of a Rocket
138 CHAPTER 7. SOME APPLICATIONS The counterintuitive answer that a supersonic aircraft must dive first in order to climb to a iven altitude in minimum time was first discovered by Walter Denham and Art
More informationProblem Set 2 Solutions
UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 125 / LeClair Sprin 2009 Problem Set 2 Solutions The followin three problems are due 20 January 2009 at the beinnin of class. 1. (H,R,&W 4.39)
More informationPhysics 18 Spring 2011 Homework 2 - Solutions Wednesday January 26, 2011
Physics 18 Sprin 011 Homework - s Wednesday January 6, 011 Make sure your name is on your homework, and please box your final answer. Because we will be ivin partial credit, be sure to attempt all the
More information1 CHAPTER 7 PROJECTILES. 7.1 No Air Resistance
CHAPTER 7 PROJECTILES 7 No Air Resistance We suppose that a particle is projected from a point O at the oriin of a coordinate system, the y-axis bein vertical and the x-axis directed alon the round The
More informationPSI AP Physics C Kinematics 2D. Multiple Choice Questions
PSI AP Physics C Kinematics D Multiple Choice Questions 1. A tennis ball is thrown off a cliff 10 m above the round with an initial horizontal velocity of 5 m/s as shown above. The time between the ball
More informationUniversity of Alabama Department of Physics and Astronomy. PH 125 / LeClair Fall Exam III Solution
University of Alabama Department of Physics and Astronomy PH 5 / LeClair Fall 07 Exam III Solution. A child throws a ball with an initial speed of 8.00 m/s at an anle of 40.0 above the horizontal. The
More informationPHYS 1114, Lecture 9, February 6 Contents:
PHYS 4, Lecture 9, February 6 Contents: Continued with projectile motion: The kicko problem in football was treated analytically, obtainin formulas for maimum heiht and rane in terms of initial speed and
More informationEnergizing Math with Engineering Applications
Enerizin Math with Enineerin Applications Understandin the Math behind Launchin a Straw-Rocket throuh the use of Simulations. Activity created by Ira Rosenthal (rosenthi@palmbeachstate.edu) as part of
More informationGet Solution of These Packages & Learn by Video Tutorials on PROJECTILE MOTION
FREE Download Study Packae from website: www.tekoclasses.com & www.mathsbysuha.com Get Solution of These Packaes & Learn by Video Tutorials on www.mathsbysuha.com. BASIC CONCEPT :. PROJECTILE PROJECTILE
More informationPrince Sultan University Physics Department First Semester 2012 /2013. PHY 105 First Major Exam Allowed Time: 60 min
Prince Sultan University Physics Department First Semester 01 /01 PHY 105 First Major Exam Allowed Time: 60 min Student Name: 1. Write your name in the specified space NOW.. Any paper without name will
More informationGeneral Relativity ASTR 2110 Sarazin. Einstein s Equation
General Relativity ASTR 2110 Sarazin Einstein s Equation Curvature of Spacetime 1. Principle of Equvalence: gravity acceleration locally 2. Acceleration curved path in spacetime In gravitational field,
More informationSmarandache Curves According to Sabban Frame on
Smarandache Curves Accordin to Sabban Frame on S Kemal Taşköprü, Murat Tosun Faculty of Arts and Sciences, Department of Mathematics Sakarya University, Sakarya 5487 TURKEY Abstract: In this paper, we
More informationPhys207: Lecture 04. Today s Agenda 3-D Kinematics Independence of x and y components Baseball projectile Shoot the monkey Uniform circular motion
Phys7: Lecture 4 Reminders All Discussion and Lab sections start meetin this week Homework is posted on course website Solutions to preious hwks will be posted Thursday mornins Today s Aenda 3-D Kinematics
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 ENGINEERING MATHEMATICS AND MECHANICS ENG-4004Y Time allowed: 2 Hours Attempt QUESTIONS 1 and 2, and ONE other question.
More informationProjectile Motion. Chin- Sung Lin STEM GARAGE SCIENCE PHYSICS
Projectile Motion Chin- Sung Lin Introduction to Projectile Motion q What is Projectile Motion? q Trajectory of a Projectile q Calculation of Projectile Motion Introduction to Projectile Motion q What
More information3.2 Projectile Motion
Motion in 2-D: Last class we were analyzing the distance in two-dimensional motion and revisited the concept of vectors, and unit-vector notation. We had our receiver run up the field then slant Northwest.
More informationGround Rules. PC1221 Fundamentals of Physics I. Position and Displacement. Average Velocity. Lectures 7 and 8 Motion in Two Dimensions
PC11 Fndamentals of Physics I Lectres 7 and 8 Motion in Two Dimensions A/Prof Tay Sen Chan 1 Grond Rles Switch off yor handphone and paer Switch off yor laptop compter and keep it No talkin while lectre
More informationMotion in Two Dimensions Sections Covered in the Text: Chapters 6 & 7, except 7.5 & 7.6
Motion in Two Dimensions Sections Covered in the Tet: Chapters 6 & 7, ecept 7.5 & 7.6 It is time to etend the definitions we developed in Note 03 to describe motion in 2D space. In doin so we shall find
More informationWhen we throw a ball :
PROJECTILE MOTION When we throw a ball : There is a constant velocity horizontal motion And there is an accelerated vertical motion These components act independently of each other PROJECTILE MOTION A
More informationPhysics 111. Lecture 7 (Walker: 4.2-5) 2D Motion Examples Projectile Motion
Physics 111 Lecture 7 (Walker: 4.-5) D Motion Eamples Projectile Motion Sept. 16, 9 -D Motion -- Constant Acceleration r r r r = v t at t v t a t y y yt y v t at r r r v = v at v = v a t v = v a t y y
More information2 General Relativity. 2.1 Curved 2D and 3D space
22 2 General Relativity The general theory of relativity (Einstein 1915) is the theory of gravity. General relativity ( Einstein s theory ) replaced the previous theory of gravity, Newton s theory. The
More informationthe equations for the motion of the particle are written as
Dynamics 4600:203 Homework 02 Due: ebruary 01, 2008 Name: Please denote your answers clearly, ie, box in, star, etc, and write neatly There are no points for small, messy, unreadable work please use lots
More informationPH Fall - Section 04 - Version A DRAFT
1. A truck (traveling in a straight line), starts from rest and accelerates to 30 m/s in 20 seconds. It cruises along at that constant speed for one minute, then brakes, coming to a stop in 25 m. Determine
More informationGraphical Vector Addition
Vectors Chapter 4 Vectors and Scalars Measured quantities can be of two types Scalar quantities: only require magnitude (and proper unit) for description. Examples: distance, speed, mass, temperature,
More informationMotion in Two Dimension (Projectile Motion)
Phsics Motion in Two Dimension (Projectile Motion) www.testprepkart.com Table of Content. Introdction.. Projectile. 3. Assmptions of projectile motion. 4. Principle of phsical independence of motions.
More informationNANO 703-Notes. Chapter 12-Reciprocal space
1 Chapter 1-Reciprocal space Conical dark-field imain We primarily use DF imain to control imae contrast, thouh STEM-DF can also ive very hih resolution, in some cases. If we have sinle crystal, a -DF
More informationThis Week. Next Week
This Week Tutorial and Test 1, in the lab (chapters 1 and 2) Next Week Experiment 1: Measurement of Lenth and Mass WileyPLUS Assinment 1 now available Due Monday, October 5 at 11:00 pm Chapters 2 & 3 28
More informationParametric Equations
Parametric Equations Suppose a cricket jumps off of the round with an initial velocity v 0 at an anle θ. If we take his initial position as the oriin, his horizontal and vertical positions follow the equations:
More informationTwo-Dimensional Motion Worksheet
Name Pd Date Two-Dimensional Motion Worksheet Because perpendicular vectors are independent of each other we can use the kinematic equations to analyze the vertical (y) and horizontal (x) components of
More informationMOTION OF A PROJECTILE
MOTION OF A PROJECTILE Today s Objectives: Students will be able to: 1. Analyze the free-flight motion of a projectile. In-Class Activities: Check Homework Reading Quiz Applications Kinematic Equations
More informationA5682: Introduction to Cosmology Course Notes. 2. General Relativity
2. General Relativity Reading: Chapter 3 (sections 3.1 and 3.2) Special Relativity Postulates of theory: 1. There is no state of absolute rest. 2. The speed of light in vacuum is constant, independent
More informationGeneral Relativity. on the frame of reference!
General Relativity Problems with special relativity What makes inertial frames special? How do you determine whether a frame is inertial? Inertial to what? Problems with gravity: In equation F = GM 1M
More informationMathematics Extension 1 Time allowed: 2 hours (plus 5 minutes reading time)
Name: Teacher: Class: FORT STREET HIGH SCHOOL 0 HIGHER SCHOOL CERTIFICATE COURSE ASSESSMENT TASK : TRIAL HSC Mathematics Extension Time allowed: hours (plus 5 minutes readin time) Syllabus Assessment Area
More informationPS 11 GeneralPhysics I for the Life Sciences
PS 11 GeneralPhysics I for the Life Sciences M E C H A N I C S I D R. B E N J A M I N C H A N A S S O C I A T E P R O F E S S O R P H Y S I C S D E P A R T M E N T N O V E M B E R 0 1 3 Definition Mechanics
More informationPhysics 351, Spring 2017, Homework #2. Due at start of class, Friday, January 27, 2017
Physics 351, Spring 2017, Homework #2. Due at start of class, Friday, January 27, 2017 Course info is at positron.hep.upenn.edu/p351 When you finish this homework, remember to visit the feedback page at
More informationFiring an Ideal Projectile
92 Chapter 13: Vector-Valued Functions and Motion in Space 13.2 Modelin Projectile Motion 921 r at time t v v cos i a j (a) v sin j Newton s second law of motion sas that the force actin on the projectile
More informationGet the frictional force from the normal force. Use dynamics to get the normal force.
. L F n µ k L =00 t µ k = 0.60 = 0 o = 050 lb F n +y +x x = sin y = cos = µf n Is the initial elocity o the car reater than 30 mph? Approach: Use conseration o enery. System: car Initial time: beore you
More informationGeneral Relativity I
General Relativity I presented by John T. Whelan The University of Texas at Brownsville whelan@phys.utb.edu LIGO Livingston SURF Lecture 2002 July 5 General Relativity Lectures I. Today (JTW): Special
More informationExam 2A Solution. 1. A baseball is thrown vertically upward and feels no air resistance. As it is rising
Exam 2A Solution 1. A baseball is thrown vertically upward and feels no air resistance. As it is risin Solution: Possible answers: A) both its momentum and its mechanical enery are conserved - incorrect.
More informationLecture: Lorentz Invariant Dynamics
Chapter 5 Lecture: Lorentz Invariant Dynamics In the preceding chapter we introduced the Minkowski metric and covariance with respect to Lorentz transformations between inertial systems. This was shown
More informationParameterization and Vector Fields
Parameterization and Vector Fields 17.1 Parameterized Curves Curves in 2 and 3-space can be represented by parametric equations. Parametric equations have the form x x(t), y y(t) in the plane and x x(t),
More informationChapter K. Oscillatory Motion. Blinn College - Physics Terry Honan. Interactive Figure
K. - Simple Harmonic Motion Chapter K Oscillatory Motion Blinn Collee - Physics 2425 - Terry Honan The Mass-Sprin System Interactive Fiure Consider a mass slidin without friction on a horizontal surface.
More informationVocabulary Preview. Oct 21 9:53 AM. Projectile Motion. An object shot through the air is called a projectile.
Projectile Trajectory Range Launch angle Vocabulary Preview Projectile Motion Projectile Motion An object shot through the air is called a projectile. A projectile can be a football, a bullet, or a drop
More informationPHY 1114: Physics I. Quick Question 1. Quick Question 2. Quick Question 3. Quick Question 4. Lecture 5: Motion in 2D
PHY 1114: Physics I Lecture 5: Motion in D Fall 01 Kenny L. Tapp Quick Question 1 A child throws a ball vertically upward at the school playground. Which one of the following quantities is (are) equal
More informationAP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:
AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented
More informationgoing vertically down, L 2 going horizontal. Observer O' outside the lift. Cut the lift wire lift accelerates wrt
PC4771 Gravitation Lectures 3&4 Einstein lift experiment Observer O in a lift, with light L 1 going vertically down, L 2 going horizontal Observer O outside the lift Cut the lift wire lift accelerates
More informationjfpr% ekuo /kez iz.ksrk ln~xq# Jh j.knksm+nklth egkjkt
Phone : 0 903 903 7779, 98930 58881 Kinematics Pae: 1 fo/u fopkjr Hkh# tu] uha kjehks dke] foifr ns[k NksM+s rqjar e/;e eu dj ';kea iq#"k fla ladyi dj] lrs foifr usd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks
More informationChapter 4. Two-Dimensional Motion
Chapter 4. Two-Dimensional Motion 09/1/003 I. Intuitive (Understanding) Review Problems. 1. If a car (object, body, truck) moves with positive velocity and negative acceleration, it means that its a) speed
More informationMotion in Two Dimensions. 1.The Position, Velocity, and Acceleration Vectors 2.Two-Dimensional Motion with Constant Acceleration 3.
Motion in Two Dimensions 1.The Position, Velocity, and Acceleration Vectors 2.Two-Dimensional Motion with Constant Acceleration 3.Projectile Motion The position of an object is described by its position
More informationProblem 2: Experiment 09 Physical Pendulum. Part One: Ruler Pendulum
Problem : Experiment 9 Physical Pendulum Part One: Ruler Pendulum The ruler has a mass m r =.159 k, a width a =.8 m, a lenth b = 1. m, and the distance from the pivot point to the center of mass is l =.479
More informationDerivatives in General Relativity
Derivatives in General Relativity One of the problems with curved space is in dealing with vectors how do you add a vector at one point in the surface of a sphere to a vector at a different point, and
More informationLinear Motion. Miroslav Mihaylov. February 13, 2014
Linear Motion Miroslav Mihaylov February 13, 2014 1 Vector components Vector A has manitude A and direction θ with respect to the horizontal. On Fiure 1 we chose the eastbound as a positive x direction
More informationExperiment 1: Simple Pendulum
COMSATS Institute of Information Technoloy, Islamabad Campus PHY-108 : Physics Lab 1 (Mechanics of Particles) Experiment 1: Simple Pendulum A simple pendulum consists of a small object (known as the bob)
More informationName: ID: Math 233 Exam 1. Page 1
Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are
More informationPhysics Chapter 3 Notes. Section 3-1: Introduction to Vectors (pages 80-83)
Physics Chapter 3 Notes Section 3-1: Introduction to Vectors (pages 80-83) We can use vectors to indicate both the magnitude of a quantity, and the direction. Vectors are often used in 2- dimensional problems.
More information10.3 Parametric Equations. 1 Math 1432 Dr. Almus
Math 1432 DAY 39 Dr. Melahat Almus almus@math.uh.edu OFFICE HOURS (212 PGH) MW12-1:30pm, F:12-1pm. If you email me, please mention the course (1432) in the subject line. Check your CASA account for Quiz
More informationExample problem: Free Fall
Example problem: Free Fall A ball is thrown from the top of a building with an initial velocity of 20.0 m/s straight upward, at an initial height of 50.0 m above the ground. The ball just misses the edge
More information3.4 The Chain Rule. F (x) = f (g(x))g (x) Alternate way of thinking about it: If y = f(u) and u = g(x) where both are differentiable functions, then
3.4 The Chain Rule To find the derivative of a function that is the composition of two functions for which we already know the derivatives, we can use the Chain Rule. The Chain Rule: Suppose F (x) = f(g(x)).
More informationPH Fall - Section 05 - Version C DRAFT
1. A truck (traveling in a straight line), starts from rest and accelerates to 30 m/s in 20 seconds. It cruises along at that constant speed for one minute, then brakes, coming to a stop in 25 m. Determine
More informationBallistics Car P3-3527
WWW.ARBORSCI.COM Ballistics Car P3-3527 BACKGROUND: The Ballistic Car demonstrates that the horizontal motion of an object is unaffected by forces which act solely in the vertical direction. It consists
More informationOval Billiards on Surfaces of Constant Curvature
Oval Billiards on Surfaces of Constant Curvature Luciano Coutinho dos Santos Sônia Pinto-de-Carvalho Abstract arxiv:1411.0236v1 [math.ds] 2 Nov 2014 In this paper we prove that the billiard problem on
More informationA Mathematical Model for the Fire-extinguishing Rocket Flight in a Turbulent Atmosphere
A Mathematical Model for the Fire-extinuishin Rocket Fliht in a Turbulent Atmosphere CRISTINA MIHAILESCU Electromecanica Ploiesti SA Soseaua Ploiesti-Tiroviste, Km 8 ROMANIA crismihailescu@yahoo.com http://www.elmec.ro
More informationMATH20411 PDEs and Vector Calculus B
MATH2411 PDEs and Vector Calculus B Dr Stefan Güttel Acknowledgement The lecture notes and other course materials are based on notes provided by Dr Catherine Powell. SECTION 1: Introctory Material MATH2411
More informationName: ID: Math 233 Exam 1. Page 1
Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are
More informationBasics of Special Relativity
Basics of Special Relativity You must understand special relativity in order to really understand general relativity. Here s a brief summary of the basic ideas and terminology of special relativity (there
More informationCHAPTER 3 MOTION IN TWO AND THREE DIMENSIONS
CHAPTER 3 MOTION IN TWO AND THREE DIMENSIONS General properties of vectors displacement vector position and velocity vectors acceleration vector equations of motion in 2- and 3-dimensions Projectile motion
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. Physics 8.01x Fall Term 2001 EXAM 1 SOLUTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01x Fall Term 2001 EXAM 1 SOLUTIONS Problem 1: We define a vertical coordinate system with positive upwards. The only forces actin
More informationPractice Problems for Final Exam
Math 1280 Spring 2016 Practice Problems for Final Exam Part 2 (Sections 6.6, 6.7, 6.8, and chapter 7) S o l u t i o n s 1. Show that the given system has a nonlinear center at the origin. ẋ = 9y 5y 5,
More information1 The Derivative and Differrentiability
1 The Derivative and Differrentiability 1.1 Derivatives and rate of change Exercise 1 Find the equation of the tangent line to f (x) = x 2 at the point (1, 1). Exercise 2 Suppose that a ball is dropped
More information11 Free vibrations: one degree of freedom
11 Free vibrations: one deree of freedom 11.1 A uniform riid disk of radius r and mass m rolls without slippin inside a circular track of radius R, as shown in the fiure. The centroidal moment of inertia
More informationSOLUTIONS TO PRACTICE PROBLEMS FOR MIDTERM I
University of California, Berkeley Physics 7A Sprin 009 (Yury Kolomensky) SOLUTIONS TO PRACTICE PROBLEMS FOR MIDTERM I Maximum score: 100 points 1. (15 points) Race Stratey Two swimmers need to et from
More informationArc Length and Curvature
Arc Length and Curvature. Last time, we saw that r(t) = cos t, sin t, t parameteried the pictured curve. (a) Find the arc length of the curve between (, 0, 0) and (, 0, π). (b) Find the unit tangent vector
More information3 The Friedmann-Robertson-Walker metric
3 The Friedmann-Robertson-Walker metric 3.1 Three dimensions The most general isotropic and homogeneous metric in three dimensions is similar to the two dimensional result of eq. (43): ( ) dr ds 2 = a
More informationChapter 4. Motion in Two Dimensions
Chapter 4 Motion in Two Dimensions Projectile Motion An object may move in both the x and y directions simultaneously. This form of two-dimensional motion we will deal with is called projectile motion.
More informationDynamics 4600:203 Homework 03 Due: February 08, 2008 Name:
Dynamics 4600:03 Homework 03 Due: ebruary 08, 008 Name: Please denote your answers clearly, i.e., bo in, star, etc., and write neatly. There are no points for small, messy, unreadable work... please use
More informationSection 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point
Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Line through P and Q approaches to the tangent line at P as Q approaches P. That is as a + h a = h gets smaller. Slope of the
More informationPHYS 100: Lecture 4 PROJECTILE MOTION. y = (v 0 /v T ) x (g/2v T2 )x 2. Velocity of Train v T. Physics 100 Lecture 4, Slide y(m)
PHYS : Lecture 4 PROJECTILE MOTION.4. Velocity of Train T y(m).8.6.4. 5 5 x(m) y ( / T ) x (/ T )x Physics Lecture 4, Slide Music Who is the Artist? A) Miles Dais B) Wynton Marsalis C) Chris Botti D) Nina
More informationcarroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general
http://pancake.uchicago.edu/ carroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. As with any major theory in physics, GR has been
More informationOscillations Equations 0. Out of the followin functions representin otion of a particle which represents SHM I) y = sinωt cosωt 3 II) y = sin ωt III) IV) 3 y = 5cos 3ωt 4 y = + ωt+ ω t a) Only IV does
More informationPROJECTILES. Launched at an Angle
PROJECTILES Launched at an Anle PROJECTILE MOTION AT AN ANGLE An bject launched int space withut mtie pwer f its wn is called a prjectile. If we nelect air resistance, the nly frce actin n a prjectile
More informationPhysics 125: Classical Physics A. 1 Practice Problems for Midterm Exam 1
Physics 125: Classical Physics A 1 Practice Problems for Midterm Exam 1 Problem 1 The Figure 1 depicts velocity as a function of time for a short run. Find: a) The acceleration at t = 5 seconds. b) The
More informationWorksheet 1.7: Introduction to Vector Functions - Position
Boise State Math 275 (Ultman) Worksheet 1.7: Introduction to Vector Functions - Position From the Toolbox (what you need from previous classes): Cartesian Coordinates: Coordinates of points in general,
More informationIntroduction to 2-Dimensional Motion
Introduction to 2-Dimensional Motion 2-Dimensional Motion! Definition: motion that occurs with both x and y components.! Example:! Playing pool.! Throwing a ball to another person.! Each dimension of the
More informationThe Perfect Cosmological Principle
Cosmoloy AS7009, 0 Lecture Outline The cosmoloical principle: Isotropy Homoeneity Bi Ban vs. Steady State cosmoloy Redshift and Hubble s law Scale factor, Hubble time, Horizon distance Olbers paradox:
More informationONLINE: MATHEMATICS EXTENSION 2 Topic 6 MECHANICS 6.3 HARMONIC MOTION
ONINE: MATHEMATICS EXTENSION Topic 6 MECHANICS 6.3 HARMONIC MOTION Vibrations or oscillations are motions that repeated more or less reularly in time. The topic is very broad and diverse and covers phenomena
More informationFrom An Apple To Black Holes Gravity in General Relativity
From An Apple To Black Holes Gravity in General Relativity Gravity as Geometry Central Idea of General Relativity Gravitational field vs magnetic field Uniqueness of trajectory in space and time Uniqueness
More informationHomework # 2. SOLUTION - We start writing Newton s second law for x and y components: F x = 0, (1) F y = mg (2) x (t) = 0 v x (t) = v 0x (3)
Physics 411 Homework # Due:..18 Mechanics I 1. A projectile is fired from the oriin of a coordinate system, in the x-y plane (x is the horizontal displacement; y, the vertical with initial velocity v =
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics Module 10 - Lecture 24 Kinematics of a particle moving on a curve Today,
More informationOSCILLATIONS
OSCIAIONS Important Points:. Simple Harmonic Motion: a) he acceleration is directly proportional to the displacement of the body from the fixed point and it is always directed towards the fixed point in
More information