...VECTOR FUNCTIONS VECTOR FUNCTIONS Motion in Space: Velocity and Acceleration In this section, we will learn about: The motion of an object using tangent and normal vectors. MOTION IN SPACE: VELOCITY AND ACCELERATION Here, we show how the ideas of tangent and normal vectors and curvature can be used in physics to study: The motion of an object, including its velocity and acceleration, along a space curve. VELOCITY AND ACCELERATION In particular, we follow in the footsteps of Newton by using these methods to derive Kepler’s First Law of planetary motion. VELOCITY Suppose a particle moves through space so that its position vector at time t is r(t). VELOCITY Vector 1 Notice from the figure that, for small values of h, the vector r(t h) r(t ) h approximates the direction of the particle moving along the curve r(t). VELOCITY Its magnitude measures the size of the displacement vector per unit time. VELOCITY The vector 1 gives the average velocity over a time interval of length h. VELOCITY VECTOR Equation 2 Its limit is the velocity vector v(t) at time t : r(t h) r(t ) v(t ) lim h 0 h r '(t ) VELOCITY VECTOR Thus, the velocity vector is also the tangent vector and points in the direction of the tangent line. SPEED The speed of the particle at time t is the magnitude of the velocity vector, that is, |v(t)|. SPEED This is appropriate because, from Equation 2 ...
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...Astronomy Notes: People & Accomplishments: Plato (428/7 B.C.E. – 328/7 B.C.E.) – Greek Philosopher, a student of Socrates, and teacher of Aristotle. Taught of the “World of Forms;” the idea that the material world (the earth) is made of nothing but imperfect copies of what was imagined to be perfect. Also, the perfect World of Forms (heavens) was where ideas, thoughts, concepts, imagination, reason, etc. exists. The seven planets ((in order; Sun, Moon, Mars, Mercury, Jupiter, Venus, and Saturn) were intangible gods; therefore, they were part of the World of Forms. Plato taught that when people acted on perfect ideas (i.e., built stuff), the outcome, in the material world, must be imperfect. [Socrates taught of metaphysics, the study of what is real versus what we think is real but isn’t.] Aristotle (384 B.C.E. – 322 B.C.E.) – Greek philosopher and mathematician and a student of Plato. Credited when the early teaching of the scientific method (questioning, predicting outcomes, classifying/ organizing data, drawing conclusions founded in logic). Aristotle taught of the Universe existing in two realms. The Terrestrial Realm consisted of all material objects. All material objects, or matter, were made of combinations of the four elements (earth, air, fire, and water). Matter was classified by common physical properties (density, hot vs. cold, wet vs. dry). Comets were thought to be atmospheric phenomena, and part of the Terrestrial Realm (changes in the tail of a comet...
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...Arturo Alcaraz (Philippines) - Instrumental in a team of scientists, who in 1967 were able to harness steam from a volcano resulting in the production of electricity. Diosdado Banatao (Philippines) - Improved computer performance throughthe development of accelerator chips, helping to make the Internet a reality. Marie Curie (Poland) - Winner of two Nobel Prizes in Chemistry and Physicsfor her studies into Radioactivity and her discoveries of Radium and Polonium. Paul Dirac (England) - An important contributor in the fields of QuantumMechanics and Electro Dynamics, Dirac was co-winner of the Nobel Prize inPhysics (1933). Albert Einstein (Germany) - Arguably needing no introduction, the most famous scientist that lived and a name that has become synonymous in popular culture with the highest intelligence. Enrico Fermi (Italy) - Heavily involved in the development of the world's first nuclear reactor and his work in induced radioactivity saw him awarded with the 1938 Nobel Prize in Physics. Vitaly Ginzburg (Russia) - One of three recipients of the 2003 Nobel inPhysics for their pioneering work in the theory of superconductors and superfluids. Christiaan Huygens (Netherlands) - Most well known for his wave theory of light, Huygens is credited with discovering the first of Saturn's moons. Werner Israel (Canada) - In 1990 Israel co-pioneered a study on black hole interiors. Ali Javan (Iran) - Born in Tehran, Ali Javan is listed as one of the top 100 living...
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...SENIOR SECONDARY COURSE PHYSICS 1 (CORE MODULES) Coordinators Dr. Oum Prakash Sharma Sh. R.S. Dass NATIONAL INSTITUTE OF OPEN SCHOOLING A-25, INSTITUTIONAL AREA, SECTOR-62, NOIDA-201301 (UP) COURSE DESIGN COMMITTEE CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Maidan Garhi, Delhi MEMBERS Prof. A.R. Verma Former Director, National Physical Laboratory, Delhi, 160, Deepali Enclave Pitampura, Delhi-34 Dr. Naresh Kumar Reader (Rtd.) Deptt. of Physics Hindu College, D.U. Dr. Oum Prakash Sharma Asstt. Director (Academic) NIOS, Delhi Prof. L.S. Kothari Prof. of Physics (Retd.) Delhi University 71, Vaishali, Delhi-11008 Dr. Vajayshree Prof. of Physics IGNOU, Maidan Garhi Delhi Sh. R.S. Dass Vice Principal (Rtd.) BRMVB, Sr. Sec. School Lajpat Nagar, New Delhi-110024 Dr. G.S. Singh Prof. of Physics IIT Roorkee Sh. K.S. Upadhyaya Principal Jawahar Navodaya Vidyalaya Rohilla Mohammadabad (U.P.) Dr. V.B. Bhatia Prof. of Physics (Retd.) Delhi University 215, Sector-21, Faridabad COURSE DEVELOPMENT TEAM CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Delhi MEMBERS Prof. V.B. Bhatia 215, Sector-21, Faridabad Prof. B.B. Tripathi Prof. of Physics (Retd.), IIT Delhi 9-A, Awadhpuri, Sarvodaya Nagar Lucknow-226016 Sh. K.S. Upadhyaya Principal Navodaya Vidyalaya Rohilla Mohammadabad, (U.P.) Dr. V.P. Shrivastava Reader (Physics) D.E.S.M., NCERT, Delhi EDITORS TEAM CHAIRMAN Prof. S.C. Garg Former Pro-Vice Chancellor IGNOU, Delhi MEMBERS Prof. B.B. Tripathi Prof...
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...g Easier! Making Everythin ™ heory tring T S Learn: • The basic concepts of this controversial theory • How string theory builds on physics concepts • The different viewpoints in the field • String theory’s physical implications Andrew Zimmerman Jones Physics Guide, About.com with Daniel Robbins, PhD in Physics Get More and Do More at Dummies.com® Start with FREE Cheat Sheets Cheat Sheets include • Checklists • Charts • Common Instructions • And Other Good Stuff! To access the Cheat Sheet created specifically for this book, go to www.dummies.com/cheatsheet/stringtheory Get Smart at Dummies.com Dummies.com makes your life easier with 1,000s of answers on everything from removing wallpaper to using the latest version of Windows. Check out our • Videos • Illustrated Articles • Step-by-Step Instructions Plus, each month you can win valuable prizes by entering our Dummies.com sweepstakes. * Want a weekly dose of Dummies? Sign up for Newsletters on • Digital Photography • Microsoft Windows & Office • Personal Finance & Investing • Health & Wellness • Computing, iPods & Cell Phones • eBay • Internet • Food, Home & Garden Find out “HOW” at Dummies.com *Sweepstakes not currently available in all countries; visit Dummies.com for official rules. String Theory FOR DUMmIES ‰ by Andrew Zimmerman Jones with Daniel Robbins, PhD in Physics String Theory For Dummies® Published by Wiley Publishing, Inc. 111 River St. Hoboken, NJ 07030-5774 www...
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