A Way to remember the Entire Unit Circle for Trigonometry
La gravitation universelle, cours de physique de seconde, 2d06ph
Pour aller plus loin : I- Le mouvement de la Lune. 1)- Le mouvement de la Lune pour un observateur terrestre. - Pour un observateur terrestre, la Lune se lève à Est et se couche à l’Ouest. - La trajectoire de la Lune dans le ciel change d’un jour à l’autre. - Le mouvement de la Lune par rapport à la Terre est complexe. - Le référentiel terrestre n’est pas adapté pour l’étude du mouvement de la Lune. - On préfère utiliser le référentiel Géocentrique. 2)- Le référentiel Géocentrique. - Le référentiel Géocentrique est un solide constitué par le centre de la Terre et des étoiles lointaines dont les positions n’ont pas varié depuis des siècles. - Le référentiel Géocentrique n’est par entraîné dans le mouvement de rotation de la Terre. - Le principe de l’inertie s’applique dans le référentiel Géocentrique. - Animation 3)- Trajectoire de la Lune. - Dans le référentiel Géocentrique, la trajectoire de la Lune est pratiquement un cercle de rayon R = 384 000 km. - Soit 60 fois le rayon de la Terre. - Énoncé :
Vector-valued function
Example[edit] A graph of the vector-valued function r(t) = <2 cos t, 4 sin t, t> indicating a range of solutions and the vector when evaluated near t = 19.5 A common example of a vector valued function is one that depends on a single real number parameter t, often representing time, producing a vector v(t) as the result. In terms of the standard unit vectors i, j, k of Cartesian 3-space, these specific type of vector-valued functions are given by expressions such as or where f(t), g(t) and h(t) are the coordinate functions of the parameter t. The vector shown in the graph to the right is the evaluation of the function near t=19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). Vector functions can also be referred to in a different notation: Properties[edit] Derivative of a three-dimensional vector function[edit] Many vector-valued functions, like scalar-valued functions, can be differentiated by simply differentiating the components in the Cartesian coordinate system. . or even
Newton's laws of motion
First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.[2][3]Second law: F = ma. The vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object.Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), first published in 1687.[4] Newton used them to explain and investigate the motion of many physical objects and systems.[5] For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. Overview Newton's first law Impulse
Loi universelle de la gravitation
Un article de Wikipédia, l'encyclopédie libre. Les satellites et les projectiles obéissent à la même loi. Expression mathématique selon Isaac Newton[modifier | modifier le code] Deux corps ponctuels de masses respectives et s'attirent avec des forces de mêmes valeurs (mais vectoriellement opposées), proportionnelles à chacune des masses, et inversement proportionnelle au carré de la distance qui les sépare. La force exercée sur le corps par le corps est vectoriellement donnée par en kilogramme (kg); d en mètre (m); en newton (N) où G est la constante gravitationnelle, elle vaut dans les unités SI, le CODATA 2010 [2] Énergie potentielle de gravitation[modifier | modifier le code] Voici le calcul menant à l'expression de l'énergie potentielle de gravitation d'un corps de masse m à une distance R d'un corps de masse M produisant le champ de gravitation : D'où : Énergie potentielle d'une sphère homogène[modifier | modifier le code] Soit un corps sphérique de rayon R et de masse volumique uniforme , on a :
Derivative
The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. The process of finding a derivative is called differentiation. Differentiation and the derivative[edit] The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y divided by x is a line. y + Δy = f(x + Δx) = m (x + Δx) + b = m x + m Δx + b = y + m Δx. It follows that Δy = m Δx. This gives an exact value for the slope of a line. Rate of change as a limit value Figure 1. Figure 2. Figure 3. Figure 4. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx as Δx becomes infinitely small. Notation[edit] Rigorous definition[edit]
Infinite [Internet Encyclopedia of Philosophy]
Working with the infinite is tricky business. Zeno’s paradoxes first alerted philosophers to this in 450 B.C.E. when he argued that a fast runner such as Achilles has an infinite number of places to reach during the pursuit of a slower runner. Since then, there has been a struggle to understand how to use the notion of infinity in a coherent manner. This article concerns the significant and controversial role that the concepts of infinity and the infinite play in the disciplines of philosophy, physical science, and mathematics. Philosophers want to know whether there is more than one coherent concept of infinity; which entities and properties are infinitely large, infinitely small, infinitely divisible, and infinitely numerous; and what arguments can justify answers one way or the other. Here are four suggested examples of these different ways to be infinite. This article also explores a variety of other questions about the infinite. Table of Contents 1. a. b. How big is infinity?
Integral
A definite integral of a function can be represented as the signed area of the region bounded by its graph. The term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written: However, the integrals discussed in this article are termed definite integrals. The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. History[edit] Pre-calculus integration[edit] The next significant advances in integral calculus did not begin to appear until the 16th century. Newton and Leibniz[edit] The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. Formalizing integrals[edit] Historical notation[edit] or
