Definition: Instantaneous Velocity Figure 3.5 A graph of instantaneous velocity as a function of time. Figure 3.6 Graph of velocity vs. time showing the tangent line at time . Using the definition of average acceleration given above, Figure 3.8 Graph of velocity as a function of time for constant. Figure 3.10 Graph of position vs. time for constant acceleration. Definition: Integral of acceleration Integral of Velocity Definition: Integral of Velocity Figure 3.13 A non-constant acceleration vs. time graph. So far we have introduced the concepts of kinematics to describe motion in one dimension; however we live in a multidimensional universe. In order to explore and describe motion in this universe, we begin by looking at examples of two-dimensional motion, of which there are many; planets orbiting a star in elliptical orbits or a projectile moving under the action of uniform gravitation are two common examples. We will now extend our definitions of position, velocity, and acceleration for an object that moves in two dimensions (in a plane) by treating each direction independently, which we can do with vector quantities by resolving each of these quantities into components. For example, our definition of velocity as the derivative of position holds for each component separately. In Cartesian coordinates, in which the directions of the unit vectors do not change from place to place, the position vector with respect to some choice of origin for the object at time is given by Consider the motion of a body that is released with an initial velocity at a height above the ground. Two paths are shown in Figure 3.14. Figure 3.15 A coordinate sketch for parabolic motion. Figure 3.16 A vector decomposition of the initial velocity Figure 3.17 The parabolic orbit Part C: Non-Constant Acceleration 3.10 Linear Friction with GravityChapter 3 Kinematics In the first place, what do we mean by time and space? It turns out that these deep philosophical questions have to be analyzed very carefully in physics, and this is not easy to do. The theory of relativity shows that our ideas of space and time are not as simple as one might imagine at first sight. However, for our present purposes, for the accuracy that we need at first, we need not be very careful about defining things precisely. Perhaps you say, “That’s a terrible thing—I learned that in science we have to define everything precisely.” We cannot define anything precisely! If we attempt to, we get into that paralysis of thought that comes to philosophers, who sit opposite each other, one saying to the other, “You don’t know what you are talking about!” The second one says. “What do you mean by know? What do you mean by talking? What do you mean by you?”, and so on. In order to be able to talk constructively, we just have to agree that we are talking roughly about the same thing. You know as much about time as you need for the present, but remember that there are some subtleties that have to be discussed; we shall discuss them later. Richard Feynman , The Feynman Lectures on Physics1 Part A: One-Dimensional Motion Introduction Kinematics is the mathematical description of motion. The term is derived from the Greek word kinema, meaning movement. In order to quantify motion, a mathematical coordinate system, called a reference frame, is used to describe space and time. Once a reference frame has been chosen, we can introduce the physical concepts of position, velocity and acceleration in a mathematically precise manner. Figure 3.1 shows a Cartesian coordinate system in one dimension with unit vector pointing in the direction of increasing ˆix-coordinate. Figure 3.1 A one-dimensional Cartesian coordinate system. 1 Richard P. Feynman, Robert B. Leighton, Matthew Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading, Massachusetts, (1963), p. 12-2. 9/5/2008 13.1 Position, Time Interval, Displacement Position Consider an object moving in one dimension. We denote the position coordinate of the center of mass of the object with respect to the choice of origin by ()xt . The position coordinate is a function of time and can be positive, zero, or negative, depending on the location of the object. The position has both direction and magnitude, and hence is a vector (Figure 3.2), ˆ() ()txt=xiG. (3.1.1) We denote the position coordinate of the center of the mass at 0t= by the symbol . The SI unit for position is the meter [m] (see Section 1.3). 0(0xxt≡=) Figure 3.2 The position vector, with reference to a chosen origin. Time Interval Consider a closed interval of time . We characterize this time interval by the difference in endpoints of the interval such that 12[, ]tt 2tt t1Δ=−. (3.1.2) The SI units for time intervals are seconds [s]. Definition: Displacement The change in position coordinate of the mass between the times and is 1t2t 21ˆ(() ()) ()ˆxtxt xtΔ≡ − ≡ΔxiGi. (3.1.3) This is called the displacement between the times and (Figure 3.3). Displacement is a vector quantity. 1t2t 9/5/2008 2Figure 3.3 The displacement vector of an object over a time interval is the vector difference between the two position vectors 3.2 Velocity When describing the motion of objects, words like “speed” and “velocity” are used in common language; however when introducing a mathematical description of motion, we need to define these terms precisely. Our procedure will be to define average quantities for finite intervals of time and then examine what happens in the limit as the time interval becomes infinitesimally small. This will lead us to the mathematical concept that velocity at an instant in time is the derivative of the position with respect to time. Definition: Average Velocity The component of the average velocity, xv , for a time interval is defined to be the displacement tΔxΔ divided by the time interval tΔ, xxvtΔ≡Δ. (3.2.1) The average velocity vector is then ˆ() ()xxtvtˆtΔ≡=ΔviiG. (3.2.2) The SI units for average velocity are meters per second 1ms−⎡⎤⋅⎣⎦. Instantaneous Velocity Consider a body moving in one direction. We denote the position coordinate of the body by ()xt , with initial position 0x at time 0t=. Consider the time interval [, . The average velocity for the interval ]tt t+ΔtΔ is the slope of the line connecting the points (, and . The slope, the rise over the run, is the
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