Linear Momentum and Collisions 261
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10. Collisions • Use Conservation of Momentum and Energy and The
10. Collisions • Use conservation of momentum and energy and the center of mass to understand collisions between two objects. • During a collision, two or more objects exert a force on one another for a short time: -F(t) F(t) Before During After • It is not necessary for the objects to touch during a collision, e.g. an asteroid flied by the earth is considered a collision because its path is changed due to the gravitational attraction of the earth. One can still use conservation of momentum and energy to analyze the collision. Impulse: During a collision, the objects exert a force on one another. This force may be complicated and change with time. However, from Newton's 3rd Law, the two objects must exert an equal and opposite force on one another. F(t) t ti tf Dt From Newton'sr 2nd Law: dp r = F (t) dt r r dp = F (t)dt r r r r tf p f - pi = Dp = ò F (t)dt ti The change in the momentum is defined as the impulse of the collision. • Impulse is a vector quantity. Impulse-Linear Momentum Theorem: In a collision, the impulse on an object is equal to the change in momentum: r r J = Dp Conservation of Linear Momentum: In a system of two or more particles that are colliding, the forces that these objects exert on one another are internal forces. These internal forces cannot change the momentum of the system. Only an external force can change the momentum. The linear momentum of a closed isolated system is conserved during a collision of objects within the system. -
Classical Mechanics
Classical Mechanics Hyoungsoon Choi Spring, 2014 Contents 1 Introduction4 1.1 Kinematics and Kinetics . .5 1.2 Kinematics: Watching Wallace and Gromit ............6 1.3 Inertia and Inertial Frame . .8 2 Newton's Laws of Motion 10 2.1 The First Law: The Law of Inertia . 10 2.2 The Second Law: The Equation of Motion . 11 2.3 The Third Law: The Law of Action and Reaction . 12 3 Laws of Conservation 14 3.1 Conservation of Momentum . 14 3.2 Conservation of Angular Momentum . 15 3.3 Conservation of Energy . 17 3.3.1 Kinetic energy . 17 3.3.2 Potential energy . 18 3.3.3 Mechanical energy conservation . 19 4 Solving Equation of Motions 20 4.1 Force-Free Motion . 21 4.2 Constant Force Motion . 22 4.2.1 Constant force motion in one dimension . 22 4.2.2 Constant force motion in two dimensions . 23 4.3 Varying Force Motion . 25 4.3.1 Drag force . 25 4.3.2 Harmonic oscillator . 29 5 Lagrangian Mechanics 30 5.1 Configuration Space . 30 5.2 Lagrangian Equations of Motion . 32 5.3 Generalized Coordinates . 34 5.4 Lagrangian Mechanics . 36 5.5 D'Alembert's Principle . 37 5.6 Conjugate Variables . 39 1 CONTENTS 2 6 Hamiltonian Mechanics 40 6.1 Legendre Transformation: From Lagrangian to Hamiltonian . 40 6.2 Hamilton's Equations . 41 6.3 Configuration Space and Phase Space . 43 6.4 Hamiltonian and Energy . 45 7 Central Force Motion 47 7.1 Conservation Laws in Central Force Field . 47 7.2 The Path Equation . -
Impulse and Momentum
Impulse and Momentum All particles with mass experience the effects of impulse and momentum. Momentum and inertia are similar concepts that describe an objects motion, however inertia describes an objects resistance to change in its velocity, and momentum refers to the magnitude and direction of it's motion. Momentum is an important parameter to consider in many situations such as braking in a car or playing a game of billiards. An object can experience both linear momentum and angular momentum. The nature of linear momentum will be explored in this module. This section will discuss momentum and impulse and the interconnection between them. We will explore how energy lost in an impact is accounted for and the relationship of momentum to collisions between two bodies. This section aims to provide a better understanding of the fundamental concept of momentum. Understanding Momentum Any body that is in motion has momentum. A force acting on a body will change its momentum. The momentum of a particle is defined as the product of the mass multiplied by the velocity of the motion. Let the variable represent momentum. ... Eq. (1) The Principle of Momentum Recall Newton's second law of motion. ... Eq. (2) This can be rewritten with accelleration as the derivate of velocity with respect to time. ... Eq. (3) If this is integrated from time to ... Eq. (4) Moving the initial momentum to the other side of the equation yields ... Eq. (5) Here, the integral in the equation is the impulse of the system; it is the force acting on the mass over a period of time to . -
12. Elastic Collisions A) Overview B) Elastic Collisions V
12. Elastic Collisions A) Overview In this unit, our focus will be on elastic collisions, namely those collisions in which the only forces that act during the collision are conservative forces. In these collisions, the sum of the kinetic energies of the objects is conserved. We will find that the description of these collisions is significantly simplified in the center of mass frame of the colliding objects. In particular, we will discover that, in this frame, the speed of each object after the collision is the same as its speed before the collision. B) Elastic Collisions In the last unit, we discussed the important topic of momentum conservation. In particular, we found that when the sum of the external forces acting on a system of particles is zero, then the total momentum of the system, defined as the vector sum of the individual momenta, will be conserved. We also determined that the kinetic energy of the system, defined to be the sum of the individual kinetic energies, is not necessarily conserved in collisions. Whether or not this energy is conserved is determined by the details of the forces that the components of the system exert on each other. In the last unit, our focus was on inelastic collisions, those collisions in which the kinetic energy of the system was not conserved. In particular non-conservative work was done by the forces that the individual objects exerted on each other during the collision. In this unit, we will look at examples in which the only forces that act during the collision are conservative forces. -
Collisions in Classical Mechanics in Terms of Mass-Momentum “Vectors” with Galilean Transformations
World Journal of Mechanics, 2020, 10, 154-165 https://www.scirp.org/journal/wjm ISSN Online: 2160-0503 ISSN Print: 2160-049X Collisions in Classical Mechanics in Terms of Mass-Momentum “Vectors” with Galilean Transformations Akihiro Ogura Laboratory of Physics, Nihon University, Matsudo, Japan How to cite this paper: Ogura, A. (2020) Abstract Collisions in Classical Mechanics in Terms of Mass-Momentum “Vectors” with Galilean We present the usefulness of mass-momentum “vectors” to analyze the colli- Transformations. World Journal of Mechan- sion problems in classical mechanics for both one and two dimensions with ics, 10, 154-165. Galilean transformations. The Galilean transformations connect the mass- https://doi.org/10.4236/wjm.2020.1010011 momentum “vectors” in the center-of-mass and the laboratory systems. We Received: August 28, 2020 show that just moving the two systems to and fro, we obtain the final states in Accepted: October 9, 2020 the laboratory systems. This gives a simple way of obtaining them, in contrast Published: October 12, 2020 with the usual way in which we have to solve the simultaneous equations. For Copyright © 2020 by author(s) and one dimensional collision, the coefficient of restitution is introduced in the Scientific Research Publishing Inc. center-of-mass system. This clearly shows the meaning of the coefficient of This work is licensed under the Creative restitution. For two dimensional collisions, we only discuss the elastic colli- Commons Attribution International sion case. We also discuss the case of which the target particle is at rest before License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ the collision. -
AIAA 19Th Fluid Dynamics, Plasma Dynamics and Lasers Conference June 8-10, 1987/Honolulu, Hawaii
AIAA-87 -1407 Electron-Cyclotron-Resonance (ECR) Plasma Acceleration J. C. Sercel Jet Propulsion Laboratory California Institute of Technology Pasadena, California AIAA 19th Fluid Dynamics, Plasma Dynamics and Lasers Conference June 8-10, 1987/Honolulu, Hawaii For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1633 Broadway, New York, NY 10019 AIAA-87-1407 ELECTRON-CYCLOTRON-RESONANCE (ECR) PLASMA ACCELERATION Joel C. Sercel* Jet Propulsion Laboratory California Institute of Technology Pasadena, California Abstract P power per unit volume, W/m3 R position vector, m A research effort directed at analytically v velocity, mls and experimentally investigating Electron U energy, J or eV Cyclotron-Resonance (ECR) plasma acceleration V electrostatic potential, volts is outlined. Relevant past research is reviewed. T temperature, Kelvin or eV The prospects for application of ECR plasma acceleration to spacecraft propulsion are described. It is shown that previously unexplained losses in converting microwave magnetic dipole moment power to directed kinetic power via ECR plasma reaction cross section, m2 acceleration can be understood in terms of diffusion of energized plasma to the physical time constant, s walls of the accelerator. It is argued that line radiation losses from electron-ion and electron SubscriPts atom inelastic collisions should be less than estimated in past research. Based on this new A acceleration understanding, the expectation now exists that B Bohm efficient ECR plasma accelerators can be e electron designed for application to high specific impulse ex excitation spacecraft propulsion. ionization summation variable refers to lowest energy level Acronyms and Abbreviations p perpendicular r relative D-He3 Deuterium Helium-Three sp space charge induced ECR Electron-Cyclotron-Resonance to t total GE General Electric JPL Jet Propulsion Laboratory LeRC Lewis Research Center I. -
6. Non-Inertial Frames
6. Non-Inertial Frames We stated, long ago, that inertial frames provide the setting for Newtonian mechanics. But what if you, one day, find yourself in a frame that is not inertial? For example, suppose that every 24 hours you happen to spin around an axis which is 2500 miles away. What would you feel? Or what if every year you spin around an axis 36 million miles away? Would that have any e↵ect on your everyday life? In this section we will discuss what Newton’s equations of motion look like in non- inertial frames. Just as there are many ways that an animal can be not a dog, so there are many ways in which a reference frame can be non-inertial. Here we will just consider one type: reference frames that rotate. We’ll start with some basic concepts. 6.1 Rotating Frames Let’s start with the inertial frame S drawn in the figure z=z with coordinate axes x, y and z.Ourgoalistounderstand the motion of particles as seen in a non-inertial frame S0, with axes x , y and z , which is rotating with respect to S. 0 0 0 y y We’ll denote the angle between the x-axis of S and the x0- axis of S as ✓.SinceS is rotating, we clearly have ✓ = ✓(t) x 0 0 θ and ✓˙ =0. 6 x Our first task is to find a way to describe the rotation of Figure 31: the axes. For this, we can use the angular velocity vector ! that we introduced in the last section to describe the motion of particles. -
Definitions and Concepts for AQA Physics a Level
Definitions and Concepts for AQA Physics A Level Topic 4: Mechanics and Materials Breaking Stress: The maximum stress that an object can withstand before failure occurs. Brittle: A brittle object will show very little strain before reaching its breaking stress. Centre of Mass: The single point through which all the mass of an object can be said to act. Conservation of Energy: Energy cannot be created or destroyed - it can only be transferred into different forms. Conservation of Momentum: The total momentum of a system before an event, must be equal to the total momentum of the system after the event, assuming no external forces act. Couple: Two equal and opposite parallel forces that act on an object through different lines of action. It has the effect of causing a rotation without translation. Density: The mass per unit volume of a material. Efficiency: The ratio of useful output to total input for a given system. Elastic Behaviour: If a material deforms with elastic behaviour, it will return to its original shape when the deforming forces are removed. The object will not be permanently deformed. Elastic Collision: A collision in which the total kinetic energy of the system before the collision is equal to the total kinetic energy of the system after the collision. Elastic Limit: The force beyond which an object will no longer deform elastically, and instead deform plastically. Beyond the elastic limit, when the deforming forces are removed, the object will not return to its original shape. Elastic Strain Energy: The energy stored in an object when it is stretched. -
Special Relativity Announcements: • Homework Solutions Will Soon Be Culearn • Homework Set 1 Returned Today
Special relativity Announcements: • Homework solutions will soon be CULearn • Homework set 1 returned today. • Homework #2 is due today. • Homework #3 is posted – due next Wed. • First midterm is 2 weeks from Christian Doppler tomorrow. (1803—1853) Today we will investigate the relativistic Doppler effect and look at momentum and energy. http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 1 Clickers Unmatched • #12ACCD73 • #3378DD96 • #3639101F • #36BC3FB5 Need to register your clickers for me to be able to associate • #39502E47 scores with you • #39BEF374 • #39CAB340 • #39CEDD2A http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 2 Clicker question 4 last lecture Set frequency to AD A spacecraft travels at speed v=0.5c relative to the Earth. It launches a missile in the forward direction at a speed of 0.5c. How fast is the missile moving relative to Earth? This actually uses the A. 0 inverse transformation: B. 0.25c C. 0.5c Have to keep signs straight. Depends on D. 0.8c which way you are transforming. Also, E. c the velocities can be positive or negative! Best way to solve these is to figure out if the speeds add or subtract and then use the appropriate formula. Since the missile if fired forward in the spacecraft frame, the spacecraft and missile velocities add in the Earth frame. http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 3 Velocity addition works with light too! A Spacecraft moving at 0.5c relative to Earth sends out a beam of light in the forward direction. What is the light velocity in the Earth frame? What about if it sends the light out in the backward direction? It works. -
Analysis of Thruster Requirements and Capabilities for Local Satellite Clusters
I I ANALYSIS OF THRUSTER REQUIREMENTS AND CAPABILITIES FOR LOCAL SATELLITE CLUSTERS G. I. Yashkot, D.E. Hastingstt I Massachusetts Institute of Technology Cambridge, Massachusetts I Abstract This paper examines the propulsive requirements necessary to maintain the relative positions of satellites orbiting in a local cluster. Formation of these large baseline arrays could allow high resolution imaging of terrestrial or I astronomical targets using techniques similar to those used for decades in radio interferometry. A key factor in the image quality is the relative positions of the individual apertures in the sparse array. The relative positions of satellites in a cluster are altered by "tidal" accelerations which are a function of the cluster baseline and orbit altitude. These accelerations must be counteracted by continuous thrusting to maintain the relative positions of the satellites. I Analysis of propulsive system requirements, limited by spacecraft power, volume, and mass constraints, indicates that specific impulses and efficiencies typical of ion engines or Hall thrusters (SPT's) are necessary to maintain large cluster baselines. In addition, required thrust to spacecraft mass ratios for reasonable size clusters are approximately I 15J..LNlkg. Finally, the ability of a proposed linear ion microthruster to meet these requirements is examined. A variation of Brophy's method is used to show that primary electron containment lengths on the order of 10 mm are I necessary to achieve those thruster characteristics. Preliminary sizing -
Bibliography of Low Energy Electron
Bibliography of Low Energy Electron Collision Cross Section Data United States Department of Commerce National Bureau of Standards Miscellaneous Publication 289 — THE NATIONAL BUREAU OF STANDARDS The National Bureau of Standards 1 provides measurement and technical information services essential to the efficiency and effectiveness of the work of the Nation's scientists and engineers. The Bureau serves also as a focal point in the Federal Government for assuring maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. To accomplish this mission, the Bureau is organized into three institutes covering broad program areas of research and services: THE INSTITUTE FOR BASIC STANDARDS . provides the central basis within the United States for a complete and consistent system of physical measurements, coordinates that system with the measurement systems of other nations, and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. This Institute comprises a series of divisions, each serving a classical subject matter area: —Applied Mathematics—Electricity—Metrology—Mechanics—Heat—Atomic Physics—Physical Chemistry—Radiation Physics—Laboratory Astrophysics 2—Radio Standards Laboratory, 2 which includes Radio Standards Physics and Radio Standards Engineering—Office of Standard Refer- ence Data. THE INSTITUTE FOR MATERIALS RESEARCH . conducts materials research and provides associated materials services including mainly reference materials and data on the properties of ma- terials. Beyond its direct interest to the Nation's scientists and engineers, this Institute yields services which are essential to the advancement of technology in industry and commerce. This Institute is or- ganized primarily by technical fields: —Analytical Chemistry—Metallurgy—Reactor Radiations—Polymers—Inorganic Materials—Cry- ogenics 2—Materials Evaluation Laboratory-— Office of Standard Reference Materials. -
The Effects of Elastic Scattering in Neutral Atom Transport D
The effects of elastic scattering in neutral atom transport D. N. Ruzic Citation: Phys. Fluids B 5, 3140 (1993); doi: 10.1063/1.860651 View online: http://dx.doi.org/10.1063/1.860651 View Table of Contents: http://pop.aip.org/resource/1/PFBPEI/v5/i9 Published by the American Institute of Physics. Related Articles Dissociation mechanisms of excited CH3X (X = Cl, Br, and I) formed via high-energy electron transfer using alkali metal targets J. Chem. Phys. 137, 184308 (2012) Efficient method for quantum calculations of molecule-molecule scattering properties in a magnetic field J. Chem. Phys. 137, 024103 (2012) Scattering resonances in slow NH3–He collisions J. Chem. Phys. 136, 074301 (2012) Accurate time dependent wave packet calculations for the N + OH reaction J. Chem. Phys. 135, 104307 (2011) The k-j-j′ vector correlation in inelastic and reactive scattering J. Chem. Phys. 135, 084305 (2011) Additional information on Phys. Fluids B Journal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors Downloaded 23 Dec 2012 to 192.17.144.173. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions The effects of elastic scattering in neutral atom transport D. N. Ruzic University of Illinois, 103South Goodwin Avenue, Urbana Illinois 61801 (Received 14 December 1992; accepted21 May 1993) Neutral atom elastic collisions are one of the dominant interactions in the edge of a high recycling diverted plasma. Starting from the quantum interatomic potentials, the scattering functions are derived for H on H ‘, H on Hz, and He on Hz in the energy range of 0.