Mathematics
Relative Motion
Relative motion refers to the measurement of the motion of one object with respect to another. In mathematics, it is often used to analyze the movement of objects in relation to a frame of reference. This concept is important for understanding the dynamics and kinematics of systems involving multiple moving parts.
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7 Key excerpts on "Relative Motion"
eBook - PDFTheoretical Mechanics for Sixth Forms
In Two Volumes
- C. Plumpton, W. A. Tomkys(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
CHAPTER V Relative Motion 5.1. Relative Position The position of a particle may be determined by reference to its displacement from an arbitrarily chosen point in space. The position thus determined is defined as the position of the particle relative to the chosen point. The frame of reference, by means of which the position of a particle is defined, is an essential feature of the definition. Thus, for example, the position of a particle in a moving railway carriage relative to another particle in the carriage may be constant, whereas the position of the first particle relative to a point on the ground outside the carriage changes with the motion of the carriage relative to the ground outside it. Because of the varied motions within the solar system, the position of a body in that system is considered relative to the sun. 5.2. Relative Velocity The velocity of a particle A relative to a particle B is defined as the rate of change of the displacement of A from B. Since the displacement of A from B is a vector, the velocity of A relative to B is also a vector. When particle A is moving with velocity u and particle B is moving with velocity v in the same direction, Fig. 5.1 (i), the component of the displacement of B from A in the direction of the motion is increasing at the rate (v— u). The component of displacement of B from A in a direc-tion at right angles to the direction of motion is constant. The velocity of B relative to A is therefore v—u in the direction of the motion. Similarly, when particle A is moving with velocity u and particle B is moving with velocity υ in the opposite direction, Fig. 5.1 (ii), the velocity of B relative to A is (v+u) in the direction of the motion of B. 106 Relative Motion 107 In each case the velocity of B relative to A is obtained by the vector addition of the velocity ofB to the reversed velocity of A.
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2022(Publication Date)
- Società Editrice Esculapio(Publisher)
Relative Motion 5.1 Introduction The choice of the reference system used to describe motion is arbitrary and the form of the equations determined using Newton’s model may depend on this choice. It is therefore sensible to wonder how to compare the description of a body motion according to observers choosing different reference systems. Any two observers placed on the origin of their reference system appear in general to move one with respect to the other. Some kind of reciprocity must exist between the mo- tions of the two observers, that is, if both observers analyse the motion of the same point- like particle, the kind of motions the two find must be related to each other by precise physical laws. The laws we find for a general case, after a short introductory argumentation about the simplest case, are transformation formulae between any two reference systems of the kine- matic quantities: coordinates, velocity and acceleration. 5.2 Relative Translational Motion Consider two observers O and ′ O who choose two different reference systems: observer O chooses a fixed, i.e. time independent, refer- ence system Oxyz, while observer ′ O choos- es a reference system ′ O ′ x ′ y ′ z moving with respect to Oxyz with velocity v ′ O = v ′ O u x and acceleration a ′ O = a ′ O u x . The coordinate axes ′ x , ′ y and ′ z are selected parallel respectively to x, y and z and unit vec- tors u ′ x , u ′ y and u ′ z have invariant di- rection while time goes by. The ′ O ′ x ′ y ′ z reference system therefore translates rigidly along the x axis with respect to the Oxyz reference system. A point-like particle placed in a point P of the space is identified by position vector r = OP in the Oxyz reference system and by vector ′ r = ′ O P in the ′ O ′ x ′ y z reference sys- tem. Furthermore the origin of the ′ O ′ x ′ y ′ z reference system in the Oxyz reference system is identified by vector r ′ O = O ′ O .
eBook - PDF- David Agmon, Paul Gluck;;;(Authors)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 3 Relative Motion and Frames of Reference Do not judge your fellow man until you are in his place. Sayings of the Fathers 3.1 Introduction Relative velocity and even more, relative acceleration are topics usually neglected in introductory textbooks. This is unfortunate, as the equivalence of all inertial frames of reference is a basic tenet in modern physics, and has also been generalized to accelerated, non-inertial systems. It is also of great importance to the learner to be able to tackle problems from different points of view, often leading him to surprisingly simple solutions and insights. The flexibility of thought induced by the maxim what you see from here may not be what you see from there is of considerable importance. At one time space was believed to be permeated by a mysterious substance called the ether. It was supposed to be at absolute rest and therefore to serve as an absolute frame of reference relative to which all motion could be measured. The famous experiment of Michelson and Morley (1896) disposed of the ether once and for all, and with it went the ability to distinguish between a state of rest and one of uniform motion with constant velocity. This led to the following important conclusions: (a) There is no such thing as absolute velocity, the velocity of any body is relative (to some frame of reference). (b) All observers moving relative to each other with constant velocity are equivalent. Every observer can argue that he is at rest and that bodies moving with respect to him are in motion. In the sequel this principle of equivalence will be generalized to accelerated systems. Equivalence is not meant to imply that different observers will describe the same event in an identical manner. Clearly, the trajectory of a moving body will look different in different reference frames. Equivalence requires that the functional form of physical laws relating dynamical variables will be identical in all inertial frames.
- Jia-shun Luo, Da-ren Wu(Authors)
- 1992(Publication Date)
- World Scientific(Publisher)
C H A P T E R I I M O T I O N A N D R E L A T I V E M O T I O N . R E L A T I V E D I F F E R E N T I A T I O N A l l material bodies i n space are moving in some way; more accurately, they move relative to one another. I n this sense, motions are all relative i n nature and the title of this chapter is rather misleading. But, in practice, we are accustomed to regard some bodies as stationary and others, moving relative to them, as i n absolute motion''. Thus, by regarding a machine tool as stationary, we used to regard the gears attached to it as in absolute motion. O f course, the machine tool is by no means fixed in space; if it is fixed on the ground, it is moving w i t h the earth relative to the sun, which is i n t u r n moving relative to other celestial bodies. Relative Motion is therefore a concept more basic than absolute motion. Actually, i n the study of gear meshing, to simplify matter, we regard both gears are i n absolute motion and deduce their motion relative to each other. But it turns out that what concerns us most is the Relative Motion of the gears i n mesh, rather than their individual absolute motions. Likewise, i n generating the teeth of a gear, it is the Relative Motion of the cutting tool and the blank that determines, together w i t h the shape of the tool, the shape of the tooth surface. I n this work, all bodies considered are supposed to be rigid and their motions, rigid motions. To study the absolute or Relative Motions of bodies, we shall imagine a right-handed frame (i .e ., a trihedral of unit vectors) be attached rigidly w i t h each of them, so that the motions concerned can be viewed as Relative Motions of the frames. I n addition, to study the geometric aspects of the moving bodies, we shall introduce a concept of relative differentiation (to be explained below), so that the ordinary analytic method can be applied to such bodies irrespective of their motions, absolute or relative. 21
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Thus, the accel- eration relative to Alex is Observers on different frames of reference that move at constant velocity rela- tive to each other will measure the same acceleration for a moving particle. 86 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS 4.7 Relative Motion IN TWO DIMENSIONS Learning Objective After reading this module, you should be able to . . . 4.7.1 Apply the relationship between a particle’s posi- tion, velocity, and acceleration as measured from two reference frames that move relative to each other at constant velocity and in two dimensions. Key Ideas ● When two frames of reference A and B are moving relative to each other at constant velocity, the velocity of a particle P as measured by an observer in frame A usually differs from that measured from frame B. The two measured velocities are related by v → PA = v → PB + v → BA , where v → BA is the velocity of B with respect to A. Both observers measure the same acceleration for the particle: a → PA = a → PB . Relative Motion in Two Dimensions Our two observers are again watching a moving particle P from the origins of refer- ence frames A and B, while B moves at a constant velocity v → BA relative to A. (The corresponding axes of these two frames remain parallel.) Figure 4.7.1 shows a cer- tain instant during the motion. At that instant, the position vector of the origin of B relative to the origin of A is r → BA . Also, the position vectors of particle P are r → PA relative to the origin of A and r → PB relative to the origin of B. From the arrangement of heads and tails of those three position vectors, we can relate the vectors with r → PA = r → PB + r → BA . (4.7.1) By taking the time derivative of this equation, we can relate the velocities v → PA and v → PB of particle P relative to our observers: v → PA = v → PB + v → BA . (4.7.2) By taking the time derivative of this relation, we can relate the accelera- tions a → PA and a → PB of the particle P relative to our observers.
eBook - PDFForces in Physics
A Historical Perspective
- Steven N. Shore(Author)
- 2008(Publication Date)
- Greenwood(Publisher)
8 THE RELATIVITY OF MOTION In the last few days I have completed one of the finest papers of my life. When you are older, I will tell you about it. —Albert Einstein to his son Hans Albert, November 4, 1915 Electromagnetic theory achieved a unification of two forces but at a price. It requi- red a medium, the ether, to support the waves that transmit the force. But this takes time and there is a delay in arrival of a signal from a source when something is changing or moving during this time interval. How do you know something is mov- ing? Remember that Aristotle used the notion of place within space or memory to recognize first change and then motion. So there the motion is known by compari- son to a previous state. But this isn’t the same question that recurs in the seven- teenth century. According to Descartes, we have to look instead at the disposition of bodies relative to which motion is known. These aren’t the same statement, although that would again divert us into metaphysics. Instead, we can look at what this means for the force concept. As Newton was at pains to explain, inertia requires two quantities, velocity and mass, to specify the quantity, momentum that is conserved. While mass is both a scalar quantity and a primitive in the dynamical principles, the momentum is not. It requires a direction and therefore, if we identify the inertial state as one moving at constant velocity, we need to know relative to what we take this motion. This is where we begin our discussion of the revolutionary developments at the start of the twentieth century: time, not only space and motion, is relative to the observer and dependent on the choice of reference frame and, consequently, has the same status as a coordinate because of its dependence on the motion of the observers. It follows that the notion of force as defined to this point in our discussions must be completely re-examined.
- 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.
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