Physics
Motion in Two Dimensions
Motion in two dimensions refers to the movement of an object in both the x and y directions simultaneously. This type of motion can be described using vectors, which include both magnitude and direction. Understanding motion in two dimensions is essential for analyzing the trajectory of projectiles, the motion of objects in circular paths, and other complex motion scenarios.
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9 Key excerpts on "Motion in Two Dimensions"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
3.2 Equations of Kinematics in Two Dimensions Motion in Two Dimensions can be described in terms of the time t and the x and y components of four vectors: the displacement, the acceleration, and the initial and final velocities. The x part of the motion occurs exactly as it would if the y part did not occur at all. Similarly, the y part of the motion occurs exactly as it would if the x part of the motion did not exist. The motion can be analyzed by treating the x and y components of the four vectors separately and realizing that the time t is the same for each component. v B 5 r B 2 r B 0 t 2 t 0 5 D r B Dt (3.1) v B 5 lim Dt B0 D r B D t (1) a B 5 v B 2 v B 0 t 2 t 0 5 Dv B D t (3.2) B a 5 lim Dt B0 D v B Dt (2) Concept Summary 65 66 Chapter 3 | Kinematics in Two Dimensions When the acceleration is constant, the x components of the displacement, the acceleration, and the initial and final velocities are related by the equations of kinematics, and so are the y components: x Component v x 5 v 0x 1 a x t (3.3a) x 5 1 2 (v 0x 1 v x )t (3.4a) x 5 v 0x t 1 1 2 a x t 2 (3.5a) v x 2 5 v 0x 2 1 2a x x (3.6a) y Component v y 5 v 0y 1 a y t (3.3b) y 5 1 2 (v 0y 1 v y )t (3.4b) y 5 v 0y t 1 1 2 a y t 2 (3.5b) v y 2 5 v 0y 2 1 2a y y (3.6b) The directions of the components of the displacement, the acceleration, and the initial and final velocities are conveyed by assigning a plus (1) or minus (2) sign to each one. 3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically down- ward. If the trajectory of the projectile is near the earth’s surface, the vertical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x 5 0 m/s 2 ), the effects of air resistance being negligible.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
We can apply the same basic equations for displacement, velocity, and acceleration we derived in Motion Along a Straight Line to describe the motion of the jets in two and three dimensions, but with some modifications—in particular, the inclusion of vectors. In this chapter we also explore two special types of Motion in Two Dimensions: projectile motion and circular motion. Last, we conclude with a discussion of relative motion. In the chapter-opening picture, each jet has a relative motion with respect Chapter 4 | Motion in Two and Three Dimensions 157 to any other jet in the group or to the people observing the air show on the ground. 4.1 | Displacement and Velocity Vectors Learning Objectives By the end of this section, you will be able to: • Calculate position vectors in a multidimensional displacement problem. • Solve for the displacement in two or three dimensions. • Calculate the velocity vector given the position vector as a function of time. • Calculate the average velocity in multiple dimensions. Displacement and velocity in two or three dimensions are straightforward extensions of the one-dimensional definitions. However, now they are vector quantities, so calculations with them have to follow the rules of vector algebra, not scalar algebra. Displacement Vector To describe motion in two and three dimensions, we must first establish a coordinate system and a convention for the axes. We generally use the coordinates x, y, and z to locate a particle at point P(x, y, z) in three dimensions. If the particle is moving, the variables x, y, and z are functions of time (t): (4.1) x = x(t) y = y(t) z = z(t). The position vector from the origin of the coordinate system to point P is r → (t). In unit vector notation, introduced in Coordinate Systems and Components of a Vector, r → (t) is (4.2) r → (t) = x(t) i ^ + y(t) j ^ + z(t) k ^ . Figure 4.2 shows the coordinate system and the vector to point P, where a particle could be located at a particular time t.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
LEARNING OBJECTIVES After reading this module, you should be able to... 3.1 Define two-dimensional kinematic variables. 3.2 Use two-dimensional kinematic equations to predict future or past values of variables. 3.3 Analyze projectile motion to predict future or past values of variables. 3.4 Apply relative velocity equations. © threespeedjones/iStockphoto CHAPTER 3 Kinematics in Two Dimensions The beautiful water fountain at the World War II Memorial in Washington, D.C., is illuminated at night. The arching streams of water follow parabolic paths whose sizes depend on the launch velocity of the water and the acceleration due to gravity, assuming that the effects of air resistance are negligible. 3.1 Displacement, Velocity, and Acceleration In Chapter 2 the concepts of displacement, velocity, and acceleration are used to describe an object moving in one dimension. There are also situations in which the motion is along a curved path that lies in a plane. Such two-dimensional motion can be described using the same concepts. In Grand Prix racing, for example, the course follows a curved road, and Figure 3.1 shows a race car at two different positions along it. These positions are identified by the vectors r → and r → 0 , which are drawn from an arbitrary coordinate origin. The displacement Δ r → of the car is the vector drawn from the initial position r → 0 at time t 0 to the final position r → at time t. The magnitude of Δ r → is the shortest distance between the two positions. In the drawing, the vectors r → 0 and Δ r → are drawn tail to head, so it is evident that r → is the vector sum of r → 0 and Δ r → . (See Sections 1.5 and 1.6 for a review of vectors and vector addition.) This means that r → = r → 0 + Δ r → , or Displacement = Δ r → = r → − r → 0 The displacement here is defined as it is in Chapter 2. Now, however, the displacement vector may lie anywhere in a plane, rather than just along a straight line. 55
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
25 © 2010 Taylor & Francis Group, LLC Motion in One Dimension This chapter addresses deriving and using the equations of motion that describe the time depen-dence of an object’s displacement, velocity, and acceleration. Relationships between displacement, velocity, and acceleration are also of importance and will be derived. This chapter starts with the basic definitions of displacement and average velocity of an object moving in one dimension. Such a simplified start will help to lead a complete set of equations of motion for an object moving along one dimension, east–west, north–south, or up and down. In the context of coordinate systems that were treated in the previous chapter, the one-dimensional motion will reduce the time and effort needed on the study of Motion in Two Dimensions, which is the subject of the next chapter. 2.1 DISPLACEMENT Motion can be defined as a continuous change in position and that change could occur in one, two, or three dimensions. As the treatment here will be limited to motions only in one dimension, one axis of the coordinate system described in Chapter 1 will suffice. Choosing this axis as the x-axis, we depict on it a point, O, which will be considered as an origin of zero coordinate. Any point on the right of the origin O will have a positive coordinate and any point on the left of O will have a negative coordinate (Figure 2.1). The displacement, denoted by Δ x, of an object as it moves from an initial position x i to a final position x f along the x-axis can be defined as the change in the object’s position along the x-axis. That is Δ x = x f – x i . (2.1) 2.1.1 S PECIAL R EMARKS The quantities in Equation 2.1 are treated as vectors. x i is the initial position vector, x f the final posi-tion vector, and Δ x the displacement vector. Accordingly, all these quantities have a direction and magnitude.
eBook - ePub- Michael M. Mansfield, Colm O'Sullivan(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
4 Motion in two and three dimensionsAIMS
- to show how, in two and three dimensions, physical quantities can be represented by mathematical entities called vectors
- to rewrite the laws of dynamics in vector form
- to study how the laws of dynamics may be applied to bodies which are constrained to move on specific paths in two and three dimensions
- to describe how the effects of friction may be included in the analysis of dynamical problems
- to study the motion of bodies which are moving on circular paths
4.1 Vector physical quantities
The material universe is a three‐dimensional world. In our investigation of the laws of motion in Chapter 3 , however, we considered only one‐dimensional motion, that is situations in which a body moves on a straight line and in which all forces applied to the body are directed along this line of motion. If a force is applied to a body in a direction other than the direction of motion the body will no longer continue to move along this line. In general, the body will travel on some path in three‐dimensional space, the detail of the trajectory depending on the magnitude and direction of the applied force at every instant. Equation (3.3) as it stands is not sufficiently general to deal with such situations, for example the motion of a pendulum bob (Figure 4.1 ) or the motion of a planet around the Sun (Figure 4.2 ). Newton's second law needs to be generalised from the simple one‐dimensional form discussed in Chapter 3 .A pendulum comprising a mass attached to the end of a string; the mass can move on a path such that the distance from the fixed end of the string remains constant.Figure 4.1A similar problem arises if two or more forces act on a body simultaneously, for example when a number of tugs are manoeuvring a large ship (Figure 4.3
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically down- ward. If the trajectory of the projectile is near the earth’s surface, the vertical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x 5 0 m/s 2 ), the effects of air resistance being negligible. There are several symmetries in projectile motion: (1) The time to reach maximum height from any vertical level is equal to the time spent returning from the maximum height to that level. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. 3.4 Relative Velocity The velocity of object A relative to object B is v B AB , and the velocity of object B relative to object C is v B BC . The velocity of A relative to C is shown in Equation 3 (note the ordering of the subscripts). While the velocity of object A relative to object B is v B AB , the velocity of B relative to A is v B AB 5 2 v B AB . v B 5 r B 2 r B 0 t 2 t 0 5 D r B Dt (3.1) v B 5 lim Dt B0 D r B D t (1) a B 5 v B 2 v B 0 t 2 t 0 5 Dv B D t (3.2) B a 5 lim Dt B0 D v B Dt (2) v B AC 5 v B AB 1 v B BC (3) x Component v x 5 v 0x 1 a x t (3.3a) x 5 1 2 (v 0x 1 v x )t (3.4a) x 5 v 0x t 1 1 2 a x t 2 (3.5a) v x 2 5 v 0x 2 1 2a x x (3.6a) y Component v y 5 v 0y 1 a y t (3.3b) y 5 1 2 (v 0y 1 v y )t (3.4b) y 5 v 0y t 1 1 2 a y t 2 (3.5b) v y 2 5 v 0y 2 1 2a y y (3.6b) FOCUS ON CONCEPTS Note to Instructors: The numbering of the questions shown here reflects the fact that they are only a representative subset of the total number that are available online. However, all of the questions are available for assignment via an online homework management program such as WileyPLUS or WebAssign.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
(4.3.2) In unit-vector notation, a → = a x ˆ i + a y ˆ j + a z k ̂ , (4.3.3) where a x = dv x /dt, a y = dv y /dt, and a z = dv z /dt. Review & Summary Projectile Motion Projectile motion is the motion of a par- ticle that is launched with an initial velocity v → 0 . During its flight, the particle’s horizontal acceleration is zero and its vertical acceleration is the free-fall acceleration ‒g. (Upward is taken to be a positive direction.) If v → 0 is expressed as a magnitude (the speed v 0 ) and an angle θ 0 (measured from the horizontal), the particle’s equations of motion along the horizontal x axis and vertical y axis are x − x 0 = ( v 0 cos θ 0 )t, (4.4.3) y − y 0 = ( v 0 sin θ 0 )t − 1 _ 2 gt 2 , (4.4.4) v y = v 0 sin θ 0 − gt, (4.4.5) v y 2 = ( v 0 sin θ 0 ) 2 − 2g(y − y 0 ). (4.4.6) The trajectory (path) of a particle in projectile motion is parabolic and is given by y = (tan θ 0 )x − gx 2 ____________ 2( v 0 cos θ 0 ) 2 , (4.4.7) if x 0 and y 0 of Eqs. 4.4.3 to 4.4.6 are zero. The particle’s horizontal range R, which is the horizontal distance from the launch point to the point at which the particle returns to the launch height, is R = v 0 2 __ g sin 2θ 0 . (4.4.8) Uniform Circular Motion If a particle travels along a circle or circular arc of radius r at constant speed v, it is said to be in uniform circular motion and has an acceleration a → of constant magnitude a = v 2 ___ r . (4.5.1) The direction of a → is toward the center of the circle or circular arc, and a → is said to be centripetal. The time for the particle to complete a circle is T = 2πr ____ v . (4.5.2) T is called the period of revolution, or simply the period, of the motion. Relative Motion 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.
eBook - PDFTheoretical Mechanics for Sixth Forms
in Two Volumes
- C. Plumpton, W. A. Tomkys(Authors)
- 2017(Publication Date)
- Pergamon(Publisher)
CHAPTER XVIII MOTION OF A PARTICLE IN TWO DIMENSIONS 18.1. A Note on Vectors In previous chapters we have distinguished between scalar quantities, such as mass, temperature, energy, which are completely specified by their magnitudes, and vector quantities, such as displacement, velocity, force, which require both magnitude and direction (and also position if they are not free vectors) for their complete specification. We have postulated the following laws in relation to vector quantities represented in magnitude and direction by straight lines: 1. AB = CD, if AB is equal in length to CD and also AB is parallel to CD in the same sense. 2. AB = —BA. 3. AB+BC = AC (the triangle of vectors). 4. AB+BC+CD+ ... + HK = AK (the polygon of vectors). 5. AB—CD = AB+(—CD). We have also discussed the parallelogram law of vectors. This law, stating that a+b = c, is illustrated in Fig. 18.1 in which vectors are denoted by single symbols in heavy bold-face type. In this notation, the positive number which represents the length of a vector a is called the modulus of a and is written either as aj or as a. A unit vector is defined as a vector with modulus 1. The unit vector whose direction is that of a is written á. The zero vector O ( null vector) is a vector whose modulus is O. 534 MOTION OF A PARTICLE IN TWO DIMENSIONS 535 Multiplication and Division of a Vector by a Real Number. Definitions 1. The result of multiplying the vector a by the positive number k is the vector ka which has the direction of the vector a and modulus ka. An immediate consequence of this definition is the relation a = a. 2. The result of multiplying the vector a by the negative number — k is the vector —ka which has the direction opposite to that of the vector a and modulus ka. 3. a : n = a/n is a vector which has the direction of a and modulus a/n. From these definitions: m(na) = mn(a) = n(ma), (m -I - h) a = ma -¤ - ha, m(a+ b) = ma -I - mb.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
71 4-7 RELATIVE Motion in Two Dimensions reference frames that move relative to each other at constant velocity and in two dimensions. Learning Objective After reading this module, you should be able to . . . 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 . Key Idea 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 4-19 shows a certain instant during the motion. At that instant, the position vector of the origin of B rela- tive 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 . (4-43) 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 . (4-44) By taking the time derivative of this relation, we can relate the accelerations a → PA and a → PB of the particle P relative to our observers. However, note that because v → BA is constant, its time derivative is zero. Thus, we get a → PA = a → PB . (4-45) As for one-dimensional motion, we have the following rule: Observers on differ- ent frames of reference that move at constant velocity relative to each other will measure the same acceleration for a moving particle. Figure 4-19 Frame B has the constant two-dimensional velocity v → BA relative to frame A. The position vector of B relative to A is r → BA . The position vectors of par- ticle P are r → PA relative to A and r → PB relative to B.
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