Physics
Kinematic Equations
Kinematic equations are a set of equations that describe the motion of objects without considering the forces causing the motion. They are used to calculate various parameters such as displacement, velocity, and acceleration of an object in motion. These equations are derived from the basic principles of motion and are fundamental in solving problems related to motion in physics.
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9 Key excerpts on "Kinematic Equations"
No longer available |Learn more- Ping YI, Jun LIU, Feng JIANG(Authors)
- 2022(Publication Date)
- EDP Sciences(Publisher)
Chapter 8 Kinetics: Equations of Motion Objectives Relate force and acceleration by equations of motion from Newton’s second law. Apply equations of motion to solve kinetic problems of a particle using different coordinate systems. Determine the mass moment of inertia of a rigid body. Derive planar kinetic equations of motion for a symmetric body undergoing planar motion. Draw free-body diagram and kinetic diagram for kinetic problems involving a rigid body or a system of connected rigid bodies. Apply planar kinetic equations of motion and appropriate kinematic relation- ships to solve kinetic problems for a rigid body or a system of connected rigid bodies. Statics deals with forces acting on particles or bodies that lead to a state of equi- librium. Then kinematics treats only the geometric aspects of the motion, without involving forces. Now we come to kinetics that relates forces acting on particles or bodies to their motions. In this chapter, based on Newton’s second law of motion, the equations of motion on different coordinate systems are first formulated and applied to solve particles’ kinetic problems. Then planar kinetic equations of motion are derived and applied to symmetric bodies undergoing planar motion. The prin- ciple of work and energy and the principle of impulse and momentum can both be derived from equations of motion, which will be discussed in the following two chapters, respectively. 8.1 Newton’s Second Law of Motion As previously stated, engineering mechanics is based on Newton’s three laws of motion. The first and third laws have been used extensively in statics. Although they are still applied in kinetics, Newton’s second law of motion forms the basis of kinetics. It states that when an unbalanced force acts on a particle, the particle has DOI: 10.1051/978-2-7598-2901-9.c008 © Science Press, EDP Sciences, 2022
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
29 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. The pilots in the United States Navy’s Blue Angels can perform high-speed maneuvers in perfect unison. They do so by controlling the displacement, velocity, and acceleration of their jet aircraft. These three concepts and the relationships among them are the focus of this chapter. 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 dimen- sion, and the next chapter treats two-dimensional motion. Dynamics deals with the effect that forces have on motion, a topic that is consid- ered in Chapter 4. Together, kinematics and dynamics form the branch of physics known as mechanics. We turn now to the first of the kine- matics concepts 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 . The length of → x 0 is the distance of the car from an arbitrarily chosen origin. At a later time the car has moved erniedecker/iStockphoto
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
36 Chapter 2 | Kinematics in One Dimension 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.
Available until 25 Jan |Learn more- Tai L. Chow(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
Also, the new frontiers of nonlinear behavior— chaos and stochastic motion—are analyzed using classical mechanics. A good grasp of classical mechanics is therefore a prerequisite for the study of the new theories of physics. This chapter will survey a number of fundamental concepts, such as velocity and accelera-tion, which are basic to our presentation in the succeeding chapters. The branch of mechanics that describes motion that does not require knowledge of its cause is called “kinematics,” and the part of mechanics that concerns the physical mechanisms that cause the motion to take place is termed “dynamics.” We shall mainly be concerned with particle dynamics; the motion of rigid bodies will be addressed in Chapter 12. A body has both mass and extent, and a particle is a body whose dimen-sions may be neglected in describing its motion. Whether we can treat the motion of a given body as that of a particle depends not only on its size but also on the conditions of the physical problem concerned. The Earth may be regarded as a particle in the context of its motion around the sun but not in a discussion of its daily rotation on its axis. 1.2 SPACE, TIME, AND COORDINATE SYSTEMS Motion involves the change in the body’s position in space as time progresses. So we shall develop classical mechanics in terms of certain notions of space and time. We shall not go into a deep philo-sophical discussion on the basic concept of space and time that is a very difficult one to comprehend or described at our level. We are concerned with the motion of bodies, and so it is only necessary for a concept of space to provide a way in which the position of such bodies can be described. We may regard space, in classical mechanics, simply as the set of possible positions that the component points of a body may occupy. Because position of a point can only be defined relative to a set of other points, position is, therefore, a relative concept.
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, 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- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Step 3. Identify exactly what needs to be determined in the problem (identify the unknowns). Step 4. Find an equation or set of equations that can help you solve the problem. Step 5. Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units. Step 6. Check the answer to see if it is reasonable: Does it make sense? 2.7 Falling Objects • An object in free-fall experiences constant acceleration if air resistance is negligible. • On Earth, all free-falling objects have an acceleration due to gravity g , which averages g = 9.80 m/s 2 . • Whether the acceleration a should be taken as +g or −g is determined by your choice of coordinate system. If you choose the upward direction as positive, a = −g = −9.80 m/s 2 is negative. In the opposite case, a = +g = 9.80 m/s 2 is positive. Since acceleration is constant, the Kinematic Equations above can be applied with the appropriate +g or −g substituted for a . • For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration. 2.8 Graphical Analysis of One-Dimensional Motion 78 Chapter 2 | Kinematics This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 • Graphs of motion can be used to analyze motion. • Graphical solutions yield identical solutions to mathematical methods for deriving motion equations. • The slope of a graph of displacement x vs. time t is velocity v . • The slope of a graph of velocity v vs. time t graph is acceleration a . • Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs. Conceptual Questions 2.1 Displacement 1. Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Specifically identify each quantity in your example.
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.
eBook - PDF- Harold Josephs, Ronald Huston(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
353 11 Generalized Dynamics: Kinematics and Kinetics 11.1 Introduction Recall in the analysis of elementary statics problems we discover, after gaining e x perience, that by making insightful choices about force directions and moment points, we can greatly simplify the analysis. Indeed, with sufficient insight, we discover that we can often obtain precisely the same number of equations as there are unknowns in the problem statement. Moreover, these equations are often uncoupled, thus producing answers with little further analysis. In Chapter 10, we found that the work–energy principle, like clever statics solution procedures, can often produce simple and direct solutions to dynamics problems. We also found, however, that while the work–energy principle is simple and direct, it is also quite restricted in its range of application. The work–energy principle leads to a single scalar equation, thus enabling the determination of a single unknown. Hence, if two or more unknowns are to be found, the work–energy principle is inadequate and is restricted to relatively simple problems. The objective of generalized dynamics is to extend the relatively simple analysis of the work–energy principle to complex dynamics problems having a number of unknowns. The intention is to equip the analyst with the means of determining unknowns with a minimal effort — as with insightful solutions of statics problems. In this chapter, we will introduce and discuss the elementary procedures of generalized dynamics. These include the concepts of generalized coordinates, partial velocities and partial angular velocities, generalized forces, and potential energy. In Chapter 12, we will use these concepts to obtain equations of motion using Kane’s equations and Lagrange’s equations. 11.2 Coordinates, Constraints, and Degrees of Freedom In the conte x t of generalized dynamics, a coordinate (or generalized coordinate ) is a parameter used to define the configuration of a mechanical system.
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