The Kinematic Equations

Unit 2 Physics: Kinamatics: M J Rhoades      . . Kinematics (from Greek κινεῖν, kinein, to move) is the branch of classical mechanics that describes the motion of bodies (objects) and systems (groups of objects) without consideration of the forces that cause the motion. · Kinematics is not to be confused with another branch of classical mechanics: analytical dynamics (the study of the relationship between the motion of objects and its causes), sometimes subdivided into kinetics (the study of the relation between external forces and motion) and statics (the study of the relations in a system at equilibrium).Kinematics also differs from dynamics as used in modern-day physics to describe time-evolution of a system. The Kinematic Equations: The goal of this first unit of The Physics Classroom has been to investigate the variety of means by which the motion of objects can be described. The variety of representations that we have investigated includes verbal representations, pictorial representations, numerical representations, and graphical representations (position-time graphs and velocity-time graphs). In Lesson 6, we will investigate the use of equations to describe and represent the motion of objects. These equations are known as kinematic equations. There are a variety of quantities associated with the motion of objects – displacement (and distance), velocity (and speed), acceleration, and time. Knowledge of each of these quantities provides descriptive information about an object’s motion. For example, if a car is known to move with a constant velocity of 22.0 m/s, north for 12.0 seconds for a northward displacement of 264 meters, then the motion of the car is fully described. And if a second car is known to accelerate from a rest position with an eastward acceleration of 3.0 m/s2 for a time of 8.0 seconds, providing a final velocity of 24 m/s, East and an eastward displacement of 96 meters, then the motion of this car is fully described. These two statements provide a complete description of the motion of an object. However, such completeness is not always known. It is often the case that only a few parameters of an object’s motion are known, while the rest are unknown. For example as you approach the stoplight, you might know that your car has a velocity of 22 m/s, east and is capable of a skidding acceleration of 8.0 m/s2, West. However you do not know the displacement that your car would experience if you were to slam on your brakes and skid to a stop; and you do not know the time required to skid to a stop. In such an instance as this, the unknown parameters...

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Random Variables And Probability Distributions

. . Random variables and probability distributions Random Variable Expected Value Variance Probability Distribution Cumulative Distribution Function Probability Density Function Discrete Random Variable Continuous Random Variable Independent Random Variables Probability-Probability (PP) Plot Quantile-Quantile (QQ) Plot Normal Distribution Poisson Distribution Binomial Distribution Geometric Distribution Uniform Distribution Central Limit Theorem . . Main Contents page | Index of all entries . Random Variable The outcome of an experiment need not be a number, for example, the outcome when a coin is tossed can be ‘heads’ or ‘tails’. However, we often want to represent outcomes as numbers. A random variable is a function that associates a unique numerical value with every outcome of an experiment. The value of the random variable will vary from trial to trial as the experiment is repeated. There are two types of random variable – discrete and continuous. A random variable has either an associated probability distribution (discrete random variable) or probability density function (continuous random variable). Examples 1.A coin is tossed ten times. The random variable X is the number of tails that are noted. X can only take the values 0, 1, …, 10, so X is a discrete random variable. 2.A light bulb is burned until it burns out. The random variable Y is its lifetime in hours. Y can take any positive real value, so Y is a continuous random variable. Expected Value The expected value (or population mean) of a random variable indicates its average or central value. It is a useful summary value (a number) of the variable’s distribution. Stating the expected value gives a general impression of the behaviour of some random variable without giving full details of its probability distribution (if it is discrete) or its probability density function (if it is continuous). Two random variables with the same expected value can have very different distributions. There are other useful descriptive measures which affect the shape of the distribution, for example variance. The expected value of a random variable X is symbolised by E(X) or µ. If X is a discrete random variable with possible values x1, x2, x3, …, xn, and p(xi) denotes P(X = xi), then the expected value of X is defined by: where the elements are summed over all values of the random variable X. If X is a continuous random variable with probability density function f(x), then the expected value of X is defined by:...

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Perfect Competitive Market

. . SUBJECT :BUSINESS ECONOMICS          Marks : 30 . . All questions are compulsory . . 1.Explain how profit maximizing output is determined in a a.Perfect Competitive market b.Monopoly c.Monopolistic market . 2.The estimated total cost function of firms is: . TC = Q0 + 2Q1 + 3Q2 + 4Q3 + 43210 . If the firms decides to produce 50 units of Q, what will be the estimated total, average and marginal costs of production? Show all the components of economic costs. . ***********************************************************************...

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Numerical Analysis Of Composite Materials

Materialenyt 1:2001, DSM (Danish Society for Materials Testing and Research) . Numerical analysis of composite materials Lauge Fuglsang Nielsen Abstract: A method is presented in this paper by which mechanical properties such as stiffness, eigenstrain/stress (e.g. shrinkage and thermal expansion), and physical properties (such as various conductivities with respect to heat, electricity, and chlorides) can be predicted for composite materials with variable geometries. A separate analysis of porous materials is made in a special section of the paper with strength estimates added to the list of composite properties considered above. The property of percolation (phase continuity) is also considered. The paper is not a ‘textbook’ in composite materials. It is a ‘users manual’ with operational introductions to the basics and running of the program COMP developed for computer analysis of composite materials. The program, which can be downloaded from the following address, is based on work previously made by the author in the area of composite materials. http://www.byg.dtu.dk/publicering/software_d.htm.          Introduction The composites considered in this paper are isotropic mixtures of two components: phase P and phase S. The amount of phase P in phase S is quantified by the so-called volume concentration defined by c = VP/(VP+VS) where volume is denoted by V. It is assumed that both phases exhibit linearity between response and gradient of potentials, which they are subjected to. For example: Mechanical stress versus deformation (Hooke’s law), heat flow versus temperature, flow of electricity versus electric potential, and diffusion of a substance versus concentration of substance. For simplicity – but also to reflect most composite problems encountered in practice – stiffness and stress results presented assume an elastic phase behaviour with Poisson’s ratios nP = nS = 0.2 (in practice nP » nS » 0.2). This means that, whenever stiffness and stress expressions are presented, they can be considered as generalized quantities, applying for any loading mode: shear, volumetric, as well as un-axial. This feature is explained in more details in a subsequent section (Composite analysis). The composite properties specifically considered in this paper are stiffness, eigenstrain (such as shrinkage and thermal expansion), and various conductivities (with respect to chloride or heat flow e.g.) as related to volume concentration, composite geometry, and phase properties: Young’s moduli EP and ES with stiffness ratio n = EP/ES, eigenstrains λP and λS, and conductivities QP and QS with conductivity ratio nQ = QP/QS. Normalized strength, S/So, of porous materials is also...

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