Mathematics
Kinematics
Kinematics is the branch of physics that deals with the motion of objects without considering the forces causing the motion. It focuses on describing the position, velocity, and acceleration of objects over time, using mathematical equations and graphs. In mathematics, kinematics is often used to study the motion of particles and systems.
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12 Key excerpts on "Kinematics"
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
An understanding of acceleration, for example, is crucial to the study of force. Our formal study of physics begins with Kinematics which is defined as the study of motion without considering its causes. The word “Kinematics” comes from a Greek term meaning motion and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion). In one-dimensional Kinematics and Two-Dimensional Kinematics we will study only the motion of a football, for example, without worrying about what forces cause or change its motion. Such considerations come in other chapters. In this chapter, we examine the simplest type of motion—namely, motion along a straight line, or one- dimensional motion. In Two-Dimensional Kinematics, we apply concepts developed here to study motion along curved paths (two- and three-dimensional motion); for example, that of a car rounding a curve. 2.1 Displacement Figure 2.2 These cyclists in Vietnam can be described by their position relative to buildings and a canal. Their motion can be described by their change in position, or displacement, in the frame of reference. (credit: Suzan Black, Fotopedia) Position In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to the Earth as a whole, while a professor’s position could be described in terms of where she is in relation to the nearby white board. (See Figure 2.3.) In other cases, we use reference frames that are not stationary but are in motion relative to the Earth.
eBook - PDF- Philip Dyke, Roger Whitworth(Authors)
- 2017(Publication Date)
- Red Globe Press(Publisher)
CHAPTER 1 Kinematics 1.1 Introduction The study of the motion of bodies requires a structured understanding of the fundamental quantities of displacement and time. This study is called kine-matics and it will provide a basis for later modelling in other branches of mechanics. From time and displacement, we derive the quantities velocity and acceleration. Displacement, velocity and acceleration are vector quantities and can be expressed in an algebraic vector form. Not surprisingly, therefore, the study of vectors is crucial to the study of Kinematics and indeed of all mechanics. We shall start our study by considering some kinematic quantities which may already be familiar. Everyday language provides us with an intuitive compre-hension of these quantities, but in some cases this familiarity can lead to serious misunderstanding, particularly where vectors are concerned. Here is an illustrative example. When a car is travelling along a road, and the speedometer reads an unchanging 30 km per hour, the driver naturally assumes that the speed is constant. The fact is that if the car is cornering, or going down or climbing up a hill, it is accelerating despite the constant speed shown on the speedometer. In the following section, we establish the concepts of displace-ment, velocity and acceleration. In particular, we clarify the distinction between speed and velocity, often used as synonyms by non-mathematicians, and the cause of the apparent contradiction of the accelerating car with its constant speedometer reading. 1.2 Definition of kinematic quantities Now we present formal definitions of displacement, distance, velocity, speed and acceleration which should help us to make a start in clearing up miscon-ceptions. Consider the fixed points P and Q , illustrated in Figure 1.1. The displacement from P to Q represented by the vector PQ ! s is the translation that is needed to move the point P to the point Q .
eBook - PDF- Bogdan Skalmierski(Author)
- 2013(Publication Date)
- Elsevier(Publisher)
CHAPTER 1 Kinematics The primary concepts in mechanics are space and time. Regarding space, we shall assume that it is Euclidean (plane), therefore an orthogonal frame of reference can be applied to it. Furthermore, it will be assumed that space is uniform and isotropic, i.e., there are no distinct loci nor preferred directions in it. With regard to time, it will be recognized as uniform, i.e., preferred instants are non-existent in it. Some parts of this book deal with problems within the purview of classical mechanics, therefore the following two hypotheses have been accepted: 1. Time is absolute, i.e. it runs identically in all frames of reference (moving and fixed). 2. The distance between two arbitrary points in space, irrespective of the frame of reference in which it is measured, is identical (it is an invariant). Kinematics is a science of motion but it is not concerned with the causes liable to induce or disturb motion. By motion we shall understand changes in time in the position of a body referred to a system treated as stationary. The position of a body or configuration is an area of Euclidean space, in which the particles of that body have been mapped one-to-one and continuously. This mapping we shall call homeomorphism. Accordingly, motion is the change of mapping in time. Note that the concept of motion is relative, since it depends on the adopted frame of reference. 1.1 Motion of a single particle We assume a right-handed frame of reference Oxyz in a space (Fig. 1.1), embracing a triplet of unit vectors, i, j, and k, which correspond to x, y, Fig. 1.1 12 Kinematics Ch. 1 and z. In a space endowed with such orientation we shall consider the motion of a single particle determined by a position vector r(t). This vector, being the function of time t, can be written as follows: r(t) = x(t)i+y(t)j+z(t)k (1) or more concisely, using the summation convention, r(t) = x`(t)e i .
- Paul Anthony Russell(Author)
- 2021(Publication Date)
- Reeds(Publisher)
The study of Kinematics concentrates on describing motion in words, numbers, diagrams, graphs, and equations. These help the engineer develop cognitive understanding about the way objects behave in the material world. The abstract realism will not be divorced from the object and forces involved; although these are Kinematics • 51 not part of the discipline, some reference to force and objects does help in shaping the engineer’s thought processes. Case A represents a body that was moving at 5 m/s due east, having its velocity changed to 12 m/s due east; the vector of each velocity is drawn from a common point; the difference between the free ends of the vectors is the change of velocity – in this case it is 7 m/s. Case B is a body with an initial velocity of 9 m/s due east, being changed to 2 m/s due west; the vector diagram shows the vector of each velocity drawn from a common point; the difference between their free ends is the change of velocity, which is 11 m/s. Case C is that of a body with an initial velocity of 6 m/s due east changed to 8 m/s due south. The vector diagram is constructed on the same principle of the two vectors drawn from a common point. The change of velocity is, as always, the difference between the free ends of the two vectors, this is, 8 6 10 2 2 + = m/s. The direction for change of velocity is S 36° 52’ W due to change in velocity taking place in the direction of the applied force, which in this case is east to south-west. In all cases, the vector diagrams are constructed by drawing the velocity vectors from a common point. This technique is called vector subtraction. Space diagrams Vector diagrams A 5 m/s 9 m/s 2 m/s 6 m/s 8 m/s 12 m/s B C N S 5 7 12 W E 9 6 8 Change of velocity 11 2 ▲ Figure 2.10 Space and vector diagrams for a change in velocity 52 • Applied Mechanics Acceleration is the rate of change of velocity; therefore, in all of these cases the value of acceleration can be obtained by dividing change of velocity by time.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
Here in Chapter 2, we begin the study of mechanics—the study of how objects move in response to forces. Before we consider forces, though, we will learn to simply describe motion, which is the subject of Kinematics. Chapter 2 deals exclusively with one- dimensional Kinematics, which is most simply defined as motion along a straight line. Pictured here is the Drop Tower thrill-ride found at Kings Dominion amusement park in Richmond, Virginia. It is the largest of its kind, with an overall height of 93 m, a drop distance of 83 m, and a maximum speed of 116 km/h. It dramatically illustrates motion along a straight line. Joel Bullock (TheCoasterCritic.com) 2 Kinematics in One Dimension 42 Position and Displacement: Vectors in One Dimension | 43 2.1 Calculate the distance and displacement when an object moves from place to place. 2.1.1 Match a vector sum to a diagram representing the graphical sum of those vectors. The world around us moves. Some motion is slow and steady, like the motion of the pas- sengers on a slow-moving escalator. Some motion is fast and unsteady, like the motion of a zigzagging gazelle being chased by a cheetah. Describing motion is the subject of Kinematics, which brings together the ideas of distance and time to form the various quan- tities that are required for a complete description of motion. Here in Chapter 2 we focus on one-dimensional motion—the simplest case. Understanding motion is a prerequisite for many topics in this text, such as the conservation of energy, thermodynamics, and special relativity. Position When specifying the position of an object, there is some uncertainty based on the object’s finite size. An automobile, for example, is about 5 m long, 2 m wide, and 2 m high, and it can assume a variety of orientations—facing north, south, and so on. Specifying a single point in space to be the “location” of the car introduces some uncertainties.
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 2 Kinematics All is influx, nothing stands still. Heraclitus Kinematics is a part of mechanics dealing with the description of motion and the functional relationships between a body's position, velocity, acceleration and time. Complex objects like a car or an airplane are treated as if they were a single mass point, called a particle, allowing one to ignore internal motions. This greatly simplifies the description of the position and related quantities. 2.1 Position and displacement Position is specified with respect to some arbitrary coordinate system, the latter always chosen to satisfy the need for convenience and simplicity. It is usual y { to work with three mutually perpendicular (Cartesian) system of axes, x,y£. For motion in a plane or along a line, two axes (x,y) or one axis (usually x) will suffice, respectively. The choice of the origin O is also a matter of convenience. The same spatial point will have different coordinates in different coordinate systems. We shall usually restrict our discussion to motion in a plane, generalization to three dimensions will be simple. The position of a point P with respect to the (x,y)-axes in the diagram may be specified in three different ways: (a) By the pair of numbers (xi,yi)» which represent the projections of the vector r on the x and y axes, respectively. (b) By the pair (r,#), where r is the distance of P from the origin and 6 is the angle between the x-axis and the line OP. (c) By the vector r, with tail at O and tip at P. The dotted line in the diagram is the time-dependent path of a particle. At time t the particle is at P located by the (position) vector /i, at a later time t 2 it is at P 2 given by the vector r 2 . The displacement vector r 2 starts from Pi and ends at P 2 , and is given by r 2 i=r 2 -r 1 as shown. We denote r 21 by Ar . Pi ^21=^2-^1 25
eBook - PDF- Harold Josephs, Ronald Huston(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
57 3 Kinematics of a Particle 3.1 Introduction Kinematics is a study of motion without regard to the cause of the motion. Often this motion occurs in three dimensions. For such cases, and even when the motion is restricted to two dimensions, it is convenient to describe the motion using vector quantities. Indeed, the principal kinematic quantities of position, velocity, and acceleration are vector quantities. In this chapter, we will study the Kinematics of particles. We will think of a “particle” as an object that is sufficiently small that it can be identified by and represented by a point. Hence, we can study the Kinematics of particles by studying the movement of points. In the next chapter, we will extend our study to rigid bodies and will think of a rigid body as being simply a collection of particles. We begin our study with a discussion of vector differentiation. 3.2 Vector Differentiation Consider a vector V whose characteristics (magnitude and direction) are dependent upon a parameter t (time). Let the functional dependence of V on t be expressed as: (3.2.1) Then, as with scalar functions, the derivative of V with respect to t is defined as: (3.2.2) The manner in which V depends upon t depends in turn upon the reference frame in which V is observed. For example, if V is fixed in a reference frame ˆ R, then in ˆ R, V is independent of t . If, however, ˆ R moves relative to a second reference frame, R, then in R V depends upon t (time). Hence, even though the rate of change of V relative to an observer in ˆ R is zero, the rate of change of V relative to an observer in R is not necessarily zero. Therefore, in general, the derivative of a vector function will depend upon the reference frame in which that derivative is calculated. Hence, to avoid ambiguity, a super-script is usually added to the derivative symbol to designate the reference frame in which V V = ( ) t d dt t t t t t V V V = + ( ) − ( ) → D Lim ∆ ∆ ∆ 0
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
CHAPTER 2 Kinematics in one dimension LEARNING OBJECTIVES After reading this module, you should be able to: 2.1 define one‐dimensional displacement 2.2 discriminate between speed and velocity 2.3 define one‐dimensional acceleration 2.4 use one‐dimensional kinematic equations to predict future or past values of variables 2.5 solve one‐dimensional kinematic problems 2.6 solve one‐dimensional free‐fall problems 2.7 predict kinematic quantities using graphical analysis. INTRODUCTION Australia holds the world record for the longest section of straight railway track: 478 kilometres of the Trans- Australian Railway that traverses the Nullarbor Plain between Sydney and Perth without a single curve. Drivers must stay vigilant for wandering kangaroos and camels as they speed across the endless kilometres of dead straight track, pressing a red ‘dead man’ switch every minute or so for safety. In this chapter we take a look at the properties of straight-line motion, such as displacement, velocity and acceleration. 1 2.1 Displacement LEARNING OBJECTIVE 2.1 Define one-dimensional displacement. There are two aspects to any motion. In a purely descriptive sense, there is the movement itself. Is it rapid or slow, for instance? Then, there is the issue of what causes the motion or what changes it, which requires that forces be considered. Kinematics deals with the concepts that are needed to describe motion, without any reference to forces. The present chapter discusses these concepts as they apply to motion in one dimension, and the next chapter treats two‐dimensional motion. Dynamics deals with the effect that forces have on motion, a topic that is considered in chapter 4. Together, Kinematics and dynamics form the branch of physics known as mechanics. We turn now to the first of the Kinematics concepts to be discussed, which is displacement. FIGURE 2.1 The displacement Δ x is a vector that points from the initial position x 0 to the final position x.
eBook - PDF- Richard C. Hill, Kirstie Plantenberg(Authors)
- 2013(Publication Date)
- SDC Publications(Publisher)
14 2.2.5) Constant acceleration equations ....................................................................... 15 2.2.6) General notes .................................................................................................... 16 2.3) ERRATIC RECTILINEAR MOTION......................................................................... 25 2.4) SOLVING RECTILINEAR PROBLEMS GRAPHICALLY ....................................... 32 CHAPTER 2 REVIEW PROBLEMS ............................................................................... 37 CHAPTER 2 PROBLEMS .............................................................................................. 41 CHAPTER 2 COMPUTER PROBLEMS ......................................................................... 48 CHAPTER 2 DESIGN PROBLEMS ................................................................................ 49 CHAPTER 2 ACTIVITIES ............................................................................................... 49 Conceptual Dynamics Kinematics: Chapter 2 – Kinematics of Particles - Rectilinear Motion 2 - 2 CHAPTER SUMMARY In this chapter, we will study Kinematics of particles. Kinematics involves the study of a body's motion without regard to the forces that generate that motion. In particular, Kinematics involves studying the relationship between displacement, velocity and acceleration. This chapter will focus on analyzing simple one-dimensional motion. The next chapter will move on to more complex two-dimensional motion. The treatment of particles precedes rigid bodies because they are simpler to analyze. A particle may be treated as a point; it has mass but no size. Therefore, we will only need to consider translational motion and not worry about rotation. Conceptual Dynamics Kinematics: Chapter 2 – Kinematics of Particles - Rectilinear Motion 2 - 3 2.1) RECTILINEAR MOTION 2.1.1) RECTILINEAR MOTION The motion of a real object with size and mass is very complex.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The pilots in the United States Navy’s Blue Angels can perform high-speed maneuvers in perfect unison. They do so by controlling the displace- ment, velocity, and acceleration of their jet aircraft. These three concepts and the relationships among them are the focus of this chapter. 2 | Kinematics in One Dimension Chapter | 2 LEARNING OBJECTIVES After reading this module, you should be able to... 2.1 | Define one-dimensional displacement. 2.2 | Discriminate between speed and velocity. 2.3 | Define one-dimensional acceleration. 2.4 | Use one-dimensional kinematic equations to predict future or past values of variables. 2.5 | Solve one-dimensional kinematic problems. 2.6 | Solve one-dimensional free-fall problems. 2.7 | Predict kinematic quantities using graphical analysis. 2.1 | Displacement There are two aspects to any motion. In a purely descriptive sense, there is the movement itself. Is it rapid or slow, for instance? Then, there is the issue of what causes the motion or what changes it, which requires that forces be considered. Kinematics deals with the concepts that are needed to describe motion, without any reference to forces. The present chapter discusses these concepts as they apply to motion in one dimension, and the next chapter treats two-dimensional motion. Dynamics deals with the effect that forces have on motion, a topic that is considered in Chapter 4. Together, Kinematics and dynamics form the branch of physics known as mechanics. We turn now to the first of the Kinematics con- cepts to be discussed, which is displacement. To describe the motion of an object, we must be able to specify the location of the object at all times, and Figure 2.1 shows how to do this for one-dimensional motion. In this drawing, the initial position of a car is indicated by the vector labeled x 0 B . The length of x 0 B is the distance of the car from an arbitrarily chosen origin. At a later time the car has moved to a new position, which is indicated by the vector x B .
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
2.5 | Applications of the Equations of Kinematics 33 2.5 | Applications of the Equations of Kinematics The equations of Kinematics can be applied to any moving object, as long as the accelera- tion of the object is constant. However, remember that each equation contains four vari- ables. Therefore, numerical values for three of the four must be available if an equation is to be used to calculate the value of the remaining variable. To avoid errors when using these equations, it helps to follow a few sensible guidelines and to be alert for a few situations that can arise during your calculations. Decide at the start which directions are to be called positive (1) and negative (2) relative to a conveniently chosen coordinate origin. This decision is arbitrary, but important, because displacement, velocity, and acceleration are vectors, and their directions must always be taken into account. In the examples that follow, the positive and negative directions will be shown in the drawings that accompany the problems. It does not matter which direc- tion is chosen to be positive. However, once the choice is made, it should not be changed during the course of the calculation. As you reason through a problem before attempting to solve it, be sure to interpret the terms “decelerating” or “deceleration” correctly, should they occur in the problem statement. These terms are the source of frequent confusion, and Conceptual Example 6 offers help in understanding them. Problem-Solving Insight Problem-Solving Insight CONCEPTUAL EXAMPLE 6 | Deceleration Versus Negative Acceleration A car is traveling along a straight road and is decelerating. Which one of the following state- ments correctly describes the car’s acceleration? (a) It must be positive. (b) It must be negative. (c) It could be positive or negative. Reasoning The term “decelerating” means that the acceleration vector points opposite to the velocity vector and indicates that the car is slowing down.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
REASONING STRATEGY Applying the Equations of Kinematics 1. Make a drawing to represent the situation being studied. A drawing helps us to see what’s happening. 2. Decide which directions are to be called positive (+) and negative (−) relative to a con- veniently chosen coordinate origin. Do not change your decision during the course of a calculation. 3. In an organized way, write down the values (with appropriate plus and minus signs) that are given for any of the five kinematic variables (x, a, υ, υ 0 , and t). Be on the alert for implicit data, such as the phrase “starts from rest,” which means that the value of the initial velocity is υ 0 = 0 m/s. The data summary tables used in the examples in the text are a good way to keep track of this information. In addition, identify the variables that you are being asked to determine. 4. Before attempting to solve a problem, verify that the given information contains values for at least three of the five kinematic variables. Once the three known variables are identified along with the desired unknown variable, the appropriate relation from Table 2.1 can be selected. Remember that the motion of two objects may be interrelated, so they may share a common variable. The fact that the motions are interrelated is an important piece of information. In such cases, data for only two variables need be specified for each object. 5. When the motion of an object is divided into segments, as in Example 8, remember that the final velocity of one segment is the initial velocity for the next segment. 6. Keep in mind that there may be two possible answers to a Kinematics problem as, for instance, in Example 7. Try to visualize the different physical situations to which the answers correspond. 2.6 Freely Falling Bodies Everyone has observed the effect of gravity as it causes objects to fall downward. In the absence of air resistance, it is found that all bodies at the same location above the earth fall vertically with the same acceleration.
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