Editing Newton's laws of motion (section)

== Prerequisites ==
Newton's laws are often stated in terms of ''point'' or ''particle'' masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each. For instance, the Earth and the Sun can both be approximated as pointlike when considering the orbit of the former around the latter, but the Earth is not pointlike when considering activities on its surface.{{refn|group=note|See, for example, Zain.<ref>{{Cite book |last=Zain |first=Samya |url=https://www.worldcat.org/oclc/1084752471 |title=Techniques of Classical Mechanics: from Lagrangian to Newtonian mechanics |date=2019 |publisher=Institute of Physics |isbn=978-0-750-32076-4 |oclc=1084752471}}</ref>{{Rp|location=1-2}} [[David Tong (physicist)|David Tong]] observes, "A particle is defined to be an object of insignificant size: e.g. an electron, a tennis ball or a planet. Obviously the validity of this statement depends on the context..."<ref>{{Cite web|last=Tong|first=David|author-link=David Tong (physicist)|date=January 2015|title=Classical Dynamics: University of Cambridge Part II Mathematical Tripos|url=http://www.damtp.cam.ac.uk/user/tong/dynamics/one.pdf|access-date=2022-02-12|website=University of Cambridge}}</ref>}}

The mathematical description of motion, or [[kinematics]], is based on the idea of specifying positions using numerical coordinates. Movement is represented by these numbers changing over time: a body's trajectory is represented by a function that assigns to each value of a time variable the values of all the position coordinates. The simplest case is one-dimensional, that is, when a body is constrained to move only along a straight line. Its position can then be given by a single number, indicating where it is relative to some chosen reference point. For example, a body might be free to slide along a track that runs left to right, and so its location can be specified by its distance from a convenient zero point, or [[Origin (mathematics)|origin]], with negative numbers indicating positions to the left and positive numbers indicating positions to the right. If the body's location as a function of time is <math>s(t)</math>, then its average velocity over the time interval from <math>t_0</math> to <math>t_1</math> is<ref name="Hughes-Hallett"/> <math display="block">\frac{\Delta s}{\Delta t} = \frac{s(t_1) - s(t_0)}{t_1 - t_0}.</math>Here, the Greek letter <math>\Delta</math> ([[Delta (letter)|delta]]) is used, per tradition, to mean "change in". A positive average velocity means that the position coordinate <math>s</math> increases over the interval in question, a negative average velocity indicates a net decrease over that interval, and an average velocity of zero means that the body ends the time interval in the same place as it began. [[Calculus]] gives the means to define an ''instantaneous'' velocity, a measure of a body's speed and direction of movement at a single moment of time, rather than over an interval. One notation for the instantaneous velocity is to replace <math>\Delta</math> with the symbol <math>d</math>, for example,<math display="block">v = \frac{ds}{dt}.</math>This denotes that the instantaneous velocity is the [[derivative]] of the position with respect to time. It can roughly be thought of as the ratio between an infinitesimally small change in position <math>ds</math> to the infinitesimally small time interval <math>dt</math> over which it occurs.<ref name="Thompson">{{cite book|author-link1=Silvanus P. Thompson |author-link2=Martin Gardner |first1=Silvanus P. |last1=Thompson |first2=Martin |last2=Gardner |year=1998 |isbn=978-0-312-18548-0 |oclc=799163595 |title=Calculus Made Easy |title-link=Calculus Made Easy |pages=84–85|publisher=Macmillan }}</ref> More carefully, the velocity and all other derivatives can be defined using the concept of a [[limit (mathematics)|limit]].<ref name="Hughes-Hallett">{{Cite book|last1=Hughes-Hallett |first1=Deborah |title=Calculus: Single and Multivariable |last2=McCallum |first2=William G. |last3=Gleason |first3=Andrew M. |last4=Connally |first4=Eric |date=2013 |publisher=Wiley |isbn=978-0-470-88861-2 |edition=6th |location=Hoboken, NJ |oclc=794034942 |display-authors=3 |author-link=Deborah Hughes Hallett |author-link2=William G. McCallum |author-link3=Andrew M. Gleason |pages=76–78}}</ref> A function <math>f(t)</math> has a limit of <math>L</math> at a given input value <math>t_0</math> if the difference between <math>f</math> and <math>L</math> can be made arbitrarily small by choosing an input sufficiently close to <math>t_0</math>. One writes, <math display="block">\lim_{t\to t_0} f(t) = L.</math>Instantaneous velocity can be defined as the limit of the average velocity as the time interval shrinks to zero:<math display="block">\frac{ds}{dt} = \lim_{\Delta t \to 0} \frac{s(t + \Delta t) - s(t)}{\Delta t}.</math> ''Acceleration'' is to velocity as velocity is to position: it is the derivative of the velocity with respect to time.{{refn|group=note|Negative acceleration includes both slowing down (when the current velocity is positive) and speeding up (when the current velocity is negative). For this and other points that students have often found difficult, see McDermott et al.<ref>{{Cite journal |last1=McDermott |first1=Lillian C. |author-link=Lillian C. McDermott |last2=Rosenquist |first2=Mark L. |last3=van Zee |first3=Emily H. |date=June 1987 |title=Student difficulties in connecting graphs and physics: Examples from kinematics |url=http://aapt.scitation.org/doi/10.1119/1.15104 |journal=[[American Journal of Physics]] |language=en |volume=55 |issue=6 |pages=503–513 |doi=10.1119/1.15104 |bibcode=1987AmJPh..55..503M |issn=0002-9505}}</ref>}} Acceleration can likewise be defined as a limit:<math display="block">a = \frac{dv}{dt} = \lim_{\Delta t \to 0}\frac{v(t+\Delta t) - v(t)}{\Delta t}.</math>Consequently, the acceleration is the ''second derivative'' of position,<ref name="Thompson"/> often written <math>\frac{d^2 s}{dt^2}</math>.

Position, when thought of as a displacement from an origin point, is a [[Vector (mathematics and physics)|vector]]: a quantity with both magnitude and direction.<ref name=":1" />{{Rp|page=1}} Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to describe motion in two, three or more dimensions. Vectors are often denoted with an arrow, as in <!-- note: math shows arrow to match text --> <math>\vec{s}</math>, or in bold typeface, such as <math>{\bf s}</math>. Often, vectors are represented visually as arrows, with the direction of the vector being the direction of the arrow, and the magnitude of the vector indicated by the length of the arrow. Numerically, a vector can be represented as a list; for example, a body's velocity vector might be <math>\mathbf{v} = (\mathrm{3~m/s}, \mathrm{4~m/s})</math>, indicating that it is moving at 3 metres per second along the horizontal axis and 4 metres per second along the vertical axis. The same motion described in a different [[coordinate system]] will be represented by different numbers, and vector algebra can be used to translate between these alternatives.<ref name=":1" />{{Rp|page=4}} 

The study of mechanics is complicated by the fact that household words like ''energy'' are used with a technical meaning.<ref>{{Cite journal |last1=Driver |first1=Rosalind |last2=Warrington |first2=Lynda |date=1985-07-01 |title=Students' use of the principle of energy conservation in problem situations |url=https://iopscience.iop.org/article/10.1088/0031-9120/20/4/308 |journal=[[Physics Education]] |volume=20 |issue=4 |pages=171–176 |doi=10.1088/0031-9120/20/4/308|bibcode=1985PhyEd..20..171D |s2cid=250781921 }}</ref><ref>{{cite journal|last=Hart |first=Christina |title=If the Sun burns you is that a force? Some definitional prerequisites for understanding Newton's laws |journal=[[Physics Education]] |volume=37 |number=3 |pages=234–238 |date=May 2002 |doi=10.1088/0031-9120/37/3/307|bibcode=2002PhyEd..37..234H }}</ref> Moreover, words which are synonymous in everyday speech are not so in physics: ''force'' is not the same as ''power'' or ''pressure'', for example, and ''mass'' has a different meaning than ''weight''.<ref>{{Cite journal |last1=Brookes |first1=David T. |last2=Etkina |first2=Eugenia |author2-link=Eugenia Etkina|date=2009-06-25 |title="Force," ontology, and language |journal=[[Physical Review Special Topics - Physics Education Research]] |language=en |volume=5 |issue=1 |pages=010110 |doi=10.1103/PhysRevSTPER.5.010110 |bibcode=2009PRPER...5a0110B |issn=1554-9178|doi-access=free }}</ref><ref name="openstax">{{cite book|url=https://openstax.org/details/books/college-physics |title=College Physics |publisher=[[OpenStax]] |year=2021 |first1=Paul Peter |last1=Urone |first2=Roger |last2=Hinrichs |first3=Kim |last3=Dirks |first4=Manjula |last4=Sharma |isbn=978-1-947172-01-2 |oclc=895896190}}</ref>{{rp|150}} The physics concept of ''force'' makes quantitative the everyday idea of a push or a pull. Forces in Newtonian mechanics are often due to strings and ropes, friction, muscle effort, gravity, and so forth. Like displacement, velocity, and acceleration, force is a vector quantity.