DOCSLIB.ORG

Spacecraft Propulsion.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spacecraft Propulsion.Pdf SPACECRAFT PROPULSION Spacecraft propulsion .................................................................................................................................... 1 Rocket technologies .................................................................................................................................. 4 Cold gas rockets .................................................................................................................................... 7 Solid propellant rockets ........................................................................................................................ 7 Liquid propellant rockets .................................................................................................................... 11 Electrothermal rockets ........................................................................................................................ 16 Electrodynamic rockets ....................................................................................................................... 16 Rocket propulsion modelling .................................................................................................................. 19 Tsiolkovsky rocket equation ............................................................................................................... 19 Delta-v budget ..................................................................................................................................... 20 Rocket exit speed ................................................................................................................................ 25 Chamber pressure equation ................................................................................................................. 27 Rocket cooling .................................................................................................................................... 29 Sub-orbital spaceflight ............................................................................................................................ 30 Sounding rockets ................................................................................................................................. 31 Air drag: ballistic coefficient .............................................................................................................. 35 Descent and reentry ................................................................................................................................. 38 Deorbiting and reentry angle ............................................................................................................... 42 Reentry phases .................................................................................................................................... 45 Landing ............................................................................................................................................... 48 Launchers ................................................................................................................................................ 49 Launcher trajectory ............................................................................................................................. 51 Launch windows and entry windows .................................................................................................. 54 Type of launchers and propellants used .............................................................................................. 54 Ariane 5 engines.................................................................................................................................. 55 Soyuz 2 engines .................................................................................................................................. 56 Vega engines ....................................................................................................................................... 58 Long March engines............................................................................................................................ 58 The Space Shuttle.................................................................................................................................... 58 Lift-off and ascent ............................................................................................................................... 60 Engines: SRB, SME+ET, OMS, RCS ................................................................................................ 64 Reentry events ..................................................................................................................................... 67 Other spacecraft propulsion systems ....................................................................................................... 69 ATV propulsion .................................................................................................................................. 69 Meteosat and AlphaBus propulsion .................................................................................................... 71 Galileo system propulsion ................................................................................................................... 72 A trip to Mars (MLS-Curiosity) .......................................................................................................... 72 Rocket emissions and other environmental effects ................................................................................. 74 SPACECRAFT PROPULSION The unique characteristic of space propulsion is the absence of a material medium (i.e. vacuum conditions), which prevents the throwing back of environmental matter for the vehicle to accelerate forward, with the following major consequences: 1. Propulsion under vacuum requires the throwing back of on-board mass. We call 'propellants' the materials to throw, and 'rocket' the engine to force the ejection; the energy to power the rocket Spacecraft propulsion 1 engine may come from inside the vehicle (e.g. combustion) or from outside (e.g. solar power); see Propulsion fundamentals, aside. 2. Propulsion under vacuum requires the carrying of both fuel and oxidiser aboard. This is a huge penalty in space propulsion, when we recall that in the burning of most fuels, stoichiometry dictates that every kilogramme of fuel requires 15 kg of air. Yes, we do not need to transport air but only oxygen, or any other oxidiser; and we might totally forget about combustion, and use other physical process to accelerate the propellants backwards to achieve propulsion. 3. Propulsion under vacuum requires the throwing of on-board mass to decelerate and brake, since there is no air drag. But, before we go on, we should specify what 'space' means in Space propulsion. By default, we understand 'outer space' where vacuum prevails, say beyond the 100 km altitude Kármán line on Earth until another planetary atmosphere is encountered; see Space environment. Besides this vast genuine outer space, we are also including under Space propulsion the atmospheric flight of space vehicles (e.g. launchers and descenders). Contrary to aircraft flight, which must be continuously propelled (sailplanes considered aside), spacecraft flight is most of the time unpowered (ballistic motion dominated by gravity forces), with some periods of propulsion based on rocket engines, i.e. on high-speed ejection of some stored mass (the propellant), accelerated by internal energy sources (e.g. chemical or nuclear reactions) or by external energy sources (e.g. natural radiation from the Sun, or radiation coming from a 'power station' spacecraft, originating in a nuclear reactor or stored solar radiation, being aimed to the recipient spacecraft in a laser or microwave beam). Another distinct feature of space propulsion is that, for the time being, there is no refuelling possibilities, and the end of a space missions is often dictated by fuel run out. We only deal here with space-related rockets (launchers and space thrusters, including sounding rockets), but rockets are not only used in spacecraft propulsion, but in some aeronautical applications (notably in missiles). Rockets for military and recreational uses (fireworks) date back to at least 13th century China. Relative motion under the outer-space vacuum must be created by ejecting own mass backwards. A rocket is the 'engine' to achieve that, although it may simply be a gas reservoir with a hole (like when an inflated balloon-toy is released open-mouth). Other means of spacecraft propulsion without mass ejection can be conceived, by making use of environmental material and electromagnetic flows in outer space (e.g. solar- sail propulsion), or the ejection of electromagnetic radiation (a flow of energy without rest-mass but with momentum), but the forces involved are too small for present applications. Space propulsion is needed for launching from a planet surface to orbit (suborbital flights included), orbit acquisition and orbit keeping against perturbations, orbit changes (including interplanetary transfers), deorbiting (descent and landing on the surface of a planet or moon), and attitude control in all phases of space flight (launch, orbiting, and deorbit). The first rocket theory is due to K.E. Tsiolkovsky (a Russian teacher, reader of Jules Verne's 'From Earth to the Moon', who developed it in 1880s and published it in 1903). He derived the rocket equation (presented Spacecraft propulsion
Load more

Recommended publications
  • The Utilization of Halo Orbit§ in Advanced Lunar Operation§
    NASA TECHNICAL NOTE NASA TN 0-6365 VI *o M *o d a THE UTILIZATION OF HALO ORBIT§ IN ADVANCED LUNAR OPERATION§ by Robert W. Fmqcbar Godddrd Spctce Flight Center P Greenbelt, Md. 20771 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. JULY 1971 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. NASA TN D-6365 5. Report Date Jul;v 1971. 6. Performing Organization Code 8. Performing Organization Report No, G-1025 10. Work Unit No. Goddard Space Flight Center 11. Contract or Grant No. Greenbelt, Maryland 20 771 13. Type of Report and Period Covered 2. Sponsoring Agency Name and Address Technical Note J National Aeronautics and Space Administration Washington, D. C. 20546 14. Sponsoring Agency Code 5. Supplementary Notes 6. Abstract Flight mechanics and control problems associated with the stationing of space- craft in halo orbits about the translunar libration point are discussed in some detail. Practical procedures for the implementation of the control techniques are described, and it is shown that these procedures can be carried out with very small AV costs. The possibility of using a relay satellite in.a halo orbit to obtain a continuous com- munications link between the earth and the far side of the moon is also discussed. Several advantages of this type of lunar far-side data link over more conventional relay-satellite systems are cited. It is shown that, with a halo relay satellite, it would be possible to continuously control an unmanned lunar roving vehicle on the moon's far side. Backside tracking of lunar orbiters could also be realized.
    [Show full text]
  • Astrodynamics
    Politecnico di Torino SEEDS SpacE Exploration and Development Systems Astrodynamics II Edition 2006 - 07 - Ver. 2.0.1 Author: Guido Colasurdo Dipartimento di Energetica Teacher: Giulio Avanzini Dipartimento di Ingegneria Aeronautica e Spaziale e-mail: [email protected] Contents 1 Two–Body Orbital Mechanics 1 1.1 BirthofAstrodynamics: Kepler’sLaws. ......... 1 1.2 Newton’sLawsofMotion ............................ ... 2 1.3 Newton’s Law of Universal Gravitation . ......... 3 1.4 The n–BodyProblem ................................. 4 1.5 Equation of Motion in the Two-Body Problem . ....... 5 1.6 PotentialEnergy ................................. ... 6 1.7 ConstantsoftheMotion . .. .. .. .. .. .. .. .. .... 7 1.8 TrajectoryEquation .............................. .... 8 1.9 ConicSections ................................... 8 1.10 Relating Energy and Semi-major Axis . ........ 9 2 Two-Dimensional Analysis of Motion 11 2.1 ReferenceFrames................................. 11 2.2 Velocity and acceleration components . ......... 12 2.3 First-Order Scalar Equations of Motion . ......... 12 2.4 PerifocalReferenceFrame . ...... 13 2.5 FlightPathAngle ................................. 14 2.6 EllipticalOrbits................................ ..... 15 2.6.1 Geometry of an Elliptical Orbit . ..... 15 2.6.2 Period of an Elliptical Orbit . ..... 16 2.7 Time–of–Flight on the Elliptical Orbit . .......... 16 2.8 Extensiontohyperbolaandparabola. ........ 18 2.9 Circular and Escape Velocity, Hyperbolic Excess Speed . .............. 18 2.10 CosmicVelocities
    [Show full text]
  • View and Print This Publication
    @ SOUTHWEST FOREST SERVICE Forest and R U. S.DEPARTMENT OF AGRICULTURE P.0. BOX 245, BERKELEY, CALIFORNIA 94701 Experime Computation of times of sunrise, sunset, and twilight in or near mountainous terrain Bill 6. Ryan Times of sunrise and sunset at specific mountain- ous locations often are important influences on for- estry operations. The change of heating of slopes and terrain at sunrise and sunset affects temperature, air density, and wind. The times of the changes in heat- ing are related to the times of reversal of slope and valley flows, surfacing of strong winds aloft, and the USDA Forest Service penetration inland of the sea breeze. The times when Research NO& PSW- 322 these meteorological reactions occur must be known 1977 if we are to predict fire behavior, smolce dispersion and trajectory, fallout patterns of airborne seeding and spraying, and prescribed burn results. ICnowledge of times of different levels of illumination, such as the beginning and ending of twilight, is necessary for scheduling operations or recreational endeavors that require natural light. The times of sunrise, sunset, and twilight at any particular location depend on such factors as latitude, longitude, time of year, elevation, and heights of the surrounding terrain. Use of the tables (such as The 1 Air Almanac1) to determine times is inconvenient Ryan, Bill C. because each table is applicable to only one location. 1977. Computation of times of sunrise, sunset, and hvilight in or near mountainous tersain. USDA Different tables are needed for each location and Forest Serv. Res. Note PSW-322, 4 p. Pacific corrections must then be made to the tables to ac- Southwest Forest and Range Exp.
    [Show full text]
  • 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.
    [Show full text]
  • Transport of Dangerous Goods
    ST/SG/AC.10/1/Rev.16 (Vol.I) Recommendations on the TRANSPORT OF DANGEROUS GOODS Model Regulations Volume I Sixteenth revised edition UNITED NATIONS New York and Geneva, 2009 NOTE The designations employed and the presentation of the material in this publication do not imply the expression of any opinion whatsoever on the part of the Secretariat of the United Nations concerning the legal status of any country, territory, city or area, or of its authorities, or concerning the delimitation of its frontiers or boundaries. ST/SG/AC.10/1/Rev.16 (Vol.I) Copyright © United Nations, 2009 All rights reserved. No part of this publication may, for sales purposes, be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, electrostatic, magnetic tape, mechanical, photocopying or otherwise, without prior permission in writing from the United Nations. UNITED NATIONS Sales No. E.09.VIII.2 ISBN 978-92-1-139136-7 (complete set of two volumes) ISSN 1014-5753 Volumes I and II not to be sold separately FOREWORD The Recommendations on the Transport of Dangerous Goods are addressed to governments and to the international organizations concerned with safety in the transport of dangerous goods. The first version, prepared by the United Nations Economic and Social Council's Committee of Experts on the Transport of Dangerous Goods, was published in 1956 (ST/ECA/43-E/CN.2/170). In response to developments in technology and the changing needs of users, they have been regularly amended and updated at succeeding sessions of the Committee of Experts pursuant to Resolution 645 G (XXIII) of 26 April 1957 of the Economic and Social Council and subsequent resolutions.
    [Show full text]
  • Delta II Icesat-2 Mission Booklet
    A United Launch Alliance (ULA) Delta II 7420-10 photon-counting laser altimeter that advances MISSION rocket will deliver the Ice, Cloud and land Eleva- technology from the first ICESat mission tion Satellite-2 (ICESat-2) spacecraft to a 250 nmi launched on a Delta II in 2003 and operated until (463 km), near-circular polar orbit. Liftoff will 2009. Our planet’s frozen and icy areas, called occur from Space Launch Complex-2 at Vanden- the cryosphere, are a key focus of NASA’s Earth berg Air Force Base, California. science research. ICESat-2 will help scientists MISSION investigate why, and how much, our cryosphere ICESat-2, with its single instrument, the is changing in a warming climate, while also Advanced Topographic Laser Altimeter System measuring heights across Earth’s temperate OVERVIEW (ATLAS), will provide scientists with height and tropical regions and take stock of the vege- measurements to create a global portrait of tation in forests worldwide. The ICESat-2 mission Earth’s third dimension, gathering data that can is implemented by NASA’s Goddard Space Flight precisely track changes of terrain including Center (GSFC). Northrop Grumman built the glaciers, sea ice, forests and more. ATLAS is a spacecraft. NASA’s Launch Services Program at Kennedy Space Center is responsible for launch management. In addition to ICESat-2, this mission includes four CubeSats which will launch from dispens- ers mounted to the Delta II second stage. The CubeSats were designed and built by UCLA, University of Central Florida, and Cal Poly. The miniaturized satellites will conduct research DELTA II For nearly 30 years, the reliable in space weather, changing electric potential Delta II rocket has been an industry and resulting discharge events on spacecraft workhorse, launching critical and damping behavior of tungsten powder in a capabilities for NASA, the Air Force Image Credit NASA’s Goddard Space Flight Center zero-gravity environment.
    [Show full text]
  • Qualification Over Ariane's Lifetime
    r bulletin 94 — may 1998 Qualification Over Ariane’s Lifetime A. González Blázquez Directorate of Launchers, ESA, Paris M. Eymard Groupe Programme CNES/Arianespace, Evry, France Introduction Similarly, the RL10 engine on the Centaur stage The primary objectives of the qualification of the Atlas launcher has been the subject of an activities performed during the operational ongoing improvement programme. About 5000 lifetime of a launcher are: tests were performed before the first flight, and – to verify the qualification status of the vehicle 4000 during the subsequent ten years. – to resolve any technical problems relating to subsystem operations on the ground or in On-going qualification activities of a similar flight. nature were started for the Ariane-3 and 4 launchers in 1986, and for Ariane-5 in 1996. Before focussing on the European family of They can be classified into two main launchers, it is perhaps informative to review categories: ‘regular’ and ‘one-off’. just one or two of the US efforts in the area of solid and liquid propulsion in order to put the Ariane-3/4 accompanying activities Ariane-related activities into context. Regular activities These activities are mainly devoted to In principle, the development programme for a launcher ends with the verification of the qualification status of the qualification phase, after which it enters operational service. In various launcher subsystems. They include the practice, however, the assessment of a launcher’s reliability is a following work packages: continuing process and qualification-type activities proceed, as an – Periodic sampling of engines: one HM7 and extension of the development programme (as is done in aeronautics), one Viking per year, tested to the limits of the over the course of the vehicle’s lifetime.
    [Show full text]
  • High-Thrust In-Space Liquid Propulsion Stage: Storable Propellants
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Institute of Transport Research:Publications Space Propulsion 2014 – ID 2968378 High-Thrust in-Space Liquid Propulsion Stage: Storable Propellants Etienne Dumont, Alexander Kopp, Carina Ludwig, Nicole Garbers DLR, Space Launcher Systems Analysis (SART), Bremen, Germany [email protected] Abstract In the frame of a project funded by ESA, a consortium led Subscripts, Abbreviations by Avio in cooperation with Snecma, Cira, and DLR is ATV Automated Transfer Vehicle performing the preliminary design of a High-Thrust in- CDF Concurrent Design Facility Space Liquid Propulsion Stage for two different types of ECSS European Cooperation on Space manned missions beyond Earth orbit. For these missions, Standardization one or two 100 ton stages are to be used to propel a Elec assy electronic assembly manned vehicle. Three different propellant combinations; EPS Etage à propergols stockables (Ariane 5’s LOx/LH2, LOx/CH4 and MON-3/MMH are being storable propellant stage) compared. GNC Guidance Navigation and Control HTS High-Thrust Stage The preliminary design of the storable variant (MON- IF Interface 3/MMH) has been performed by DLR. The Aestus II ISS International Space Station engine with a large nozzle expansion ratio has been LEO Low Earth Orbit chosen as baseline. A first iteration has demonstrated, that LEOP Launch and Early Orbit Phase it indeed provides the best performance for the storable LH2 Liquid Hydrogen propellant combination, when considering all engines LOx Liquid Oxygen available today or which may be available in a short- to MLI Multi-Layer Insulation medium term.
    [Show full text]
  • 1 the Merits of Cold Gas Micropropulsion in State-Of
    IAC-02-S.2.07 THE MERITS OF COLD GAS MICROPROPULSION IN STATE-OF-THE-ART SPACE MISSIONS Hugo Nguyen, Johan Köhler and Lars Stenmark The Ångström Space Technology Centre, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden. [email protected] Fax: +46 18 471 3572 Abstract: Cold gas micropropulsion is a sound 1. INTRODUCTION choice for space missions that require extreme For Attitude and Orbit Control System (AOCS) stabilisation, pointing precision or contamination- requirements on extreme stabilisation, pointing free operation. The use of forces in the micronewton precision, contamination-free operation is the range for spacecraft operations has been identified as most important for many missions. Examples a mission-critical item in several demanding space include DARWIN, LISA, SIM, NGST, and those systems currently under development. which have optical or other equipments pointing Cold gas micropropulsion systems share merits with exactly to a certain object or in a certain direction. traditional cold gas systems in being simple in Other obviously desirable properties include design, clean, safe, and robust. They do not generate design simplicity, cleanliness, safety, robustness, net charge to the spacecraft, and typically operate on low-power operation, no net charge generation to low-power. The minute size is suitable not only for the craft, together with low mass, and a wide inclusion on high-performance nanosatellites but dynamic range. Cold gas micropropulsion has also for high-demanding future space missions of those merits and in the present paper, larger sizes. technological solutions will be treated By using differently sized nozzles in parallel emphatically, at the same time the mentioned systems the dynamic range of a cold gas merits will be brought out clearly.
    [Show full text]
  • Investigation of Condensed and Early Stage Gas Phase Hypergolic Reactions Jacob Daniel Dennis Purdue University
    Purdue University Purdue e-Pubs Open Access Dissertations Theses and Dissertations Fall 2014 Investigation of condensed and early stage gas phase hypergolic reactions Jacob Daniel Dennis Purdue University Follow this and additional works at: https://docs.lib.purdue.edu/open_access_dissertations Part of the Propulsion and Power Commons Recommended Citation Dennis, Jacob Daniel, "Investigation of condensed and early stage gas phase hypergolic reactions" (2014). Open Access Dissertations. 256. https://docs.lib.purdue.edu/open_access_dissertations/256 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. i INVESTIGATION OF CONDENSED AND EARLY STAGE GAS PHASE HYPERGOLIC REACTIONS A Dissertation Submitted to the Faculty of Purdue University by Jacob Daniel Dennis In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2014 Purdue University West Lafayette, Indiana ii To my parents, Jay and Susan Dennis, who have always pushed me to be the person they know I am capable of being. Also to my wife, Claresta Dennis, who not only tolerated me but suffered along with me throughout graduate school. I love you and am so proud of you! iii ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor, Dr. Timothée Pourpoint, for guiding me over the past four years and helping me become the researcher that I am today. In addition I would like to thank the rest of my PhD Committee for the insight and guidance. I would also like to acknowledge the help provided by my fellow graduate students who spent time with me in the lab: Travis Kubal, Yair Solomon, Robb Janesheski, Jordan Forness, Jonathan Chrzanowski, Jared Willits, and Jason Gabl.
    [Show full text]
  • The Physics of a Car Collision by Andrew Zimmerman Jones, Thoughtco.Com on 09.10.19 Word Count 947 Level MAX
    The physics of a car collision By Andrew Zimmerman Jones, ThoughtCo.com on 09.10.19 Word Count 947 Level MAX Image 1. A crash test dummy sits inside a Toyota Corolla during the 2017 North American International Auto Show in Detroit, Michigan. Crash test dummies are used to predict the injuries that a human might sustain in a car crash. Photo by: Jim Watson/AFP/Getty Images During a car crash, energy is transferred from the vehicle to whatever it hits, be it another vehicle or a stationary object. This transfer of energy, depending on variables that alter states of motion, can cause injuries and damage cars and property. The object that was struck will either absorb the energy thrust upon it or possibly transfer that energy back to the vehicle that struck it. Focusing on the distinction between force and energy can help explain the physics involved. Force: Colliding With A Wall Car crashes are clear examples of how Newton's Laws of Motion work. His first law of motion, also referred to as the law of inertia, asserts that an object in motion will stay in motion unless an external force acts upon it. Conversely, if an object is at rest, it will remain at rest until an unbalanced force acts upon it. Consider a situation in which car A collides with a static, unbreakable wall. The situation begins with car A traveling at a velocity (v) and, upon colliding with the wall, ending with a velocity of 0. The force of this situation is defined by Newton's second law of motion, which uses the equation of This article is available at 5 reading levels at https://newsela.com.
    [Show full text]
  • Rocket Propulsion Fundamentals 2
    https://ntrs.nasa.gov/search.jsp?R=20140002716 2019-08-29T14:36:45+00:00Z Liquid Propulsion Systems – Evolution & Advancements Launch Vehicle Propulsion & Systems LPTC Liquid Propulsion Technical Committee Rick Ballard Liquid Engine Systems Lead SLS Liquid Engines Office NASA / MSFC All rights reserved. No part of this publication may be reproduced, distributed, or transmitted, unless for course participation and to a paid course student, in any form or by any means, or stored in a database or retrieval system, without the prior written permission of AIAA and/or course instructor. Contact the American Institute of Aeronautics and Astronautics, Professional Development Program, Suite 500, 1801 Alexander Bell Drive, Reston, VA 20191-4344 Modules 1. Rocket Propulsion Fundamentals 2. LRE Applications 3. Liquid Propellants 4. Engine Power Cycles 5. Engine Components Module 1: Rocket Propulsion TOPICS Fundamentals • Thrust • Specific Impulse • Mixture Ratio • Isp vs. MR • Density vs. Isp • Propellant Mass vs. Volume Warning: Contents deal with math, • Area Ratio physics and thermodynamics. Be afraid…be very afraid… Terms A Area a Acceleration F Force (thrust) g Gravity constant (32.2 ft/sec2) I Impulse m Mass P Pressure Subscripts t Time a Ambient T Temperature c Chamber e Exit V Velocity o Initial state r Reaction ∆ Delta / Difference s Stagnation sp Specific ε Area Ratio t Throat or Total γ Ratio of specific heats Thrust (1/3) Rocket thrust can be explained using Newton’s 2nd and 3rd laws of motion. 2nd Law: a force applied to a body is equal to the mass of the body and its acceleration in the direction of the force.
    [Show full text]