Editing Kinematics

{{short description|Branch of physics describing the motion of objects without considering forces}}
{{redirect|Kinematic|the music group|Kinematic (band)}}
{{Classical mechanics|expanded=branches}}

In [[physics]], '''kinematics''' studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics.

Kinematics is concerned with systems of specification of objects' positions and velocities and mathematical transformations between such systems. These systems may be rectangular like [[Cartesian coordinate system|cartesian]], [[Curvilinear coordinates]] like [[polar coordinates]] or other systems. The object trajectories may be specified with respect to other objects which may themselve be in motion relative to a standard reference. Rotating systems may also be used.

Numerous practical problems in kinematics involve constraints, such as mechanical linkages, ropes, or rolling disks.

== Overview ==

'''Kinematics''' is a subfield of [[physics]] and [[mathematics]], developed in [[classical mechanics]], that describes the [[motion]] of points, [[Physical object|bodies]] (objects), and systems of bodies (groups of objects) without considering the [[force]]s that cause them to move.<ref name="Whittaker">{{cite book |title=[[A Treatise on the Analytical Dynamics of Particles and Rigid Bodies]]|author=Edmund Taylor Whittaker|author-link=E. T. Whittaker |at=Chapter 1 |year=1904 |publisher=Cambridge University Press |isbn=0-521-35883-3}}</ref><ref name=Beggs>{{cite book |title=Kinematics |author=Joseph Stiles Beggs |page=1 |url=https://books.google.com/books?id=y6iJ1NIYSmgC |isbn=0-89116-355-7 |year=1983 |publisher=Taylor & Francis}}</ref><ref name=Wright>{{cite book |title=Elements of Mechanics Including Kinematics, Kinetics and Statics|author=Thomas Wallace Wright |url=https://books.google.com/books?id=-LwLAAAAYAAJ |at=Chapter 1 |year=1896 |publisher=E and FN Spon}}</ref> 
The study of how forces act on bodies falls within [[kinetics (physics)|''kinetics'']] or [[Dynamics (mechanics)|''dynamics'']] (including [[analytical dynamics]]), not kinematics.

Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure [[mathematics]] since it can be studied without considering the mass of a body or the forces acting upon it.<ref>{{cite book |title=Engineering Mechanics: Dynamics |author=Russell C. Hibbeler |chapter=Kinematics and kinetics of a particle |chapter-url=https://books.google.com/books?id=tOFRjXB-XvMC&pg=PA298 |page=298 |isbn=978-0-13-607791-6 |year=2009 |edition=12th |publisher=Prentice Hall}}</ref><ref>{{cite book |title=Dynamics of Multibody Systems |author=Ahmed A. Shabana |chapter=Reference kinematics |chapter-url=https://books.google.com/books?id=zxuG-l7J5rgC&pg=PA28 |edition=2nd |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-54411-5}}</ref><ref>{{cite book |title=Mechanical Systems, Classical Models: Particle Mechanics |chapter=Kinematics |page=287 |chapter-url=https://books.google.com/books?id=k4H2AjWh9qQC&pg=PA287 |author=P. P. Teodorescu |isbn=978-1-4020-5441-9 |year=2007 |publisher=Springer}}.</ref> A kinematics problem begins by describing the geometry of the system and declaring the [[initial conditions]] of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined.

Another way to describe kinematics is as the specification of the possible states of a physical system. Dynamics then describes the evolution of a system through such states. [[Robert Spekkens]] argues that this division cannot be empirically tests and thus has no physical basis.<ref>{{Cite book |last=Spekkens |first=Robert W. |chapter=The Paradigm of Kinematics and Dynamics Must Yield to Causal Structure |series=The Frontiers Collection |date=2015 |editor-last=Aguirre |editor-first=Anthony |editor2-last=Foster |editor2-first=Brendan |editor3-last=Merali |editor3-first=Zeeya |title=Questioning the Foundations of Physics |chapter-url=https://link.springer.com/10.1007/978-3-319-13045-3_2 |language=en |location=Cham |publisher=Springer International Publishing |pages=5–16 |doi=10.1007/978-3-319-13045-3_2 |isbn=978-3-319-13044-6}}</ref>

Kinematics is used in [[astrophysics]] to describe the motion of [[celestial bodies]] and collections of such bodies. In [[mechanical engineering]], [[robotics]], and [[biomechanics]],<ref name="Biewener">{{cite book |title=Animal Locomotion |url=https://books.google.com/books?id=yMaN9pk8QJAC |author=A. Biewener |isbn=019850022X |publisher=Oxford University Press |year=2003}}</ref> kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an [[engine]], a [[robot kinematics|robotic arm]] or the [[human skeleton]].

[[Geometric transformation]]s, including called [[rigid transformation]]s, are used to describe the movement of components in a [[mechanical system]], simplifying the derivation of the equations of motion. They are also central to [[Lagrangian mechanics|dynamic analysis]].

[[Robot kinematics|Kinematic analysis]] is the process of measuring the kinematic [[Physical quantity|quantities]] used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given [[Mechanism (engineering)|mechanism]] and, working in reverse, using [[Burmester theory|kinematic synthesis]] to design a mechanism for a desired range of motion.<ref name="McCarthy2010">J. M. McCarthy and G. S. Soh, 2010, [https://books.google.com/books?id=jv9mQyjRIw4C&pg=PA231 ''Geometric Design of Linkages,''] Springer, New York.</ref> <!-- I am sure that these are the best examples: The movement of a crane and the oscillations of a piston in an engine are both simple kinematic systems. The crane is a type of open kinematic chain, while the piston is part of a closed [[four-bar linkage]]. --> In addition, kinematics applies [[algebraic geometry]] to the study of the [[mechanical advantage]] of a [[mechanical system]] or mechanism.

[[Relativistic mechanics#Relativistic kinematics|Relativistic_kinematics]] applies the [[special theory of relativity]] to the geometry of object motion. The topics include [[time dilation]], [[length contraction]] and the [[Lorentz transformation]].<ref>{{Cite book |last1=Kleppner |first1=Daniel |url=https://www.cambridge.org/core/product/identifier/9781139013963/type/book |title=An Introduction to Mechanics |last2=Kolenkow |first2=Robert |date=2013-11-18 |publisher=Cambridge University Press |isbn=978-0-521-19811-0 |edition=2 |doi=10.1017/cbo9781139013963}}</ref>{{rp|loc=12.8}} The kinematics of relativity operates in a [[spacetime]] geometry where spatial points are augmented with a time coordinate to form [[four-vectors|4-vectors]].<ref>{{Cite book |last=Zee |first=Anthony |title=Einstein Gravity in a Nutshell |date=2013 |publisher=Princeton University Press |isbn=978-0-691-14558-7 |edition=1st |series=In a Nutshell Series |location=Princeton}}</ref>{{rp|221}}

[[Werner Heisenberg]] reinterpreted classical kinetics for quantum systems in his 1925 paper [[Umdeutung paper|"On the quantum-theoretical reinterpretation of kinematical and mechanical relationships"]].<ref>Heisenberg, Werner. "Quantum-theoretical re-interpretation of kinematic and mechanical relations." Z. Phys 33 (1925): 879-893.</ref> Dirac noted the similarity in structure between Heisenberg's formulations and classical [[Poisson brackets]].<ref>{{Cite book |last1=Mehra |first1=J |title=The Historical Development of Quantum Theory. |last2=Rechenberg |first2=H |publisher=Springer |year=2000}}</ref>{{rp|143}} In a follow up paper in 1927 Heisenberg showed that classical kinematic notions like velocity and energy are valid in quantum mechanics, but pairs of conjugate kinematic and dynamic quantities cannot be simultaneously measure, a  result he called indeterminacy but which became known as the [[uncertainty principle]].<ref>{{Cite book |last=Heisenberg |first=Werner |title=Quantum Theory and Measurement |date=1983 |publisher=Princeton University Press |isbn=978-1-4008-5455-4 |editor-last=Wheeler |editor-first=John Archibald |series=Princeton series in physics |location=Princeton, New Jersey |chapter=The Physical Content of Quantum Kinematics and Mechanics |editor-last2=Zurek |editor-first2=Wojciech Hubert}}</ref>

== Etymology ==
The term kinematic is the English version of [[André-Marie Ampère|A.M. Ampère]]'s ''cinématique'',<ref>{{cite book |last=Ampère |first=André-Marie |author-link=André-Marie Ampère |title=Essai sur la Philosophie des Sciences |year=1834 |publisher=Chez Bachelier |url=https://archive.org/details/bub_gb_j4QPAAAAQAAJ}}</ref> which he constructed from the [[Ancient Greek language|Greek]] {{lang|grc|κίνημα}} ''kinema'' ("movement, motion"), itself derived from {{lang|grc|κινεῖν}} ''kinein'' ("to move").<ref>{{cite book |last=Merz |first=John |title=A History of European Thought in the Nineteenth Century |publisher=Blackwood, London |year=1903 |pages=[https://archive.org/details/historyofeuropea02merzuoft/page/5 5] |url=https://archive.org/details/historyofeuropea02merzuoft}}</ref><ref name="Bottema">{{cite book |title=Theoretical Kinematics |at=preface, p. 5 |url=https://books.google.com/books?id=f8I4yGVi9ocC|author=O. Bottema & B. Roth |isbn=0-486-66346-9 |publisher=Dover Publications |year=1990}}</ref>

Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and from the Greek {{lang|grc|γρᾰ́φω}} ''grapho'' ("to write").<ref>{{OEtymD|cinema}}</ref>

== Kinematics of a particle trajectory in a non-rotating frame of reference ==
[[File:Kinematics.svg|thumb|300px|Kinematic quantities of a classical particle: mass ''m'', position '''r''', velocity '''v''', acceleration '''a'''.]]
{{multiple image
|align = vertical
|width1 = 100
|image1 = Position vector plane polar coords.svg
|caption1 = Position vector '''r''', always points radially from the origin.
|width2 = 150
|image2 = Velocity vector plane polar coords.svg
|caption2 = Velocity vector '''v''', always tangent to the path of motion.
|width3 = 200
|image3 = Acceleration vector plane polar coords.svg
|caption3 = Acceleration vector '''a''', not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
|footer = Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2-d space, but a plane in any higher dimension.}}

Particle kinematics is the study of the trajectory of particles. The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50&nbsp;m south from your home, where the coordinate frame is centered at your home, such that east is in the direction of the ''x''-axis and north is in the direction of the ''y''-axis, then the coordinate vector to the base of the tower is '''r''' = (0&nbsp;m, −50&nbsp;m, 0&nbsp;m). If the tower is 50&nbsp;m high, and this height is measured along the ''z''-axis, then the coordinate vector to the top of the tower is '''r''' = (0&nbsp;m, −50&nbsp;m, 50&nbsp;m).

In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to a reference frame.

The position vector of a particle is a [[Euclidean vector|vector]] drawn from the origin of the [[reference frame]] to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position vector <math>{\bf r}</math> can be expressed as
<math display="block">\mathbf r = (x,y,z) = x\hat\mathbf x + y\hat\mathbf y + z\hat\mathbf z,</math>
where <math>x</math>, <math>y</math>, and <math>z</math> are the [[Cartesian coordinates]] and <math> \hat\mathbf x</math>, <math>\hat\mathbf y</math> and <math>\hat\mathbf z</math> are the [[unit vectors]] along the <math>x</math>, <math>y</math>, and <math>z</math> coordinate axes, respectively. The magnitude of the position vector <math>\left|\mathbf r\right|</math> gives the distance between the point <math>\mathbf r</math> and the origin.
<math display="block">|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}.</math>
The [[direction cosine]]s of the position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector.

The ''trajectory'' of a particle is a vector function of time, <math>\mathbf{r}(t)</math>, which defines the curve traced by the moving particle, given by
<math display="block"> \mathbf r(t) = x(t)\hat\mathbf x + y(t) \hat\mathbf y +z(t) \hat\mathbf z,</math>
where <math>x(t)</math>, <math>y(t)</math>, and <math>z(t)</math> describe each coordinate of the particle's position as a function of time.

[[File:Distancedisplacement.svg|thumb|300px|right|The distance travelled is always greater than or equal to the displacement.]]

===Velocity and speed===
The [[velocity]] of a particle is a vector quantity that describes the ''direction'' as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle ([[Displacement (geometry)|displacement]]) by the time interval. This ratio is called the [[average velocity]] over that time interval and is defined as<math display="block"> \mathbf\bar v = \frac{\Delta \mathbf r}{\Delta t} = \frac{\Delta x}{\Delta t}\hat\mathbf x + \frac{\Delta y}{\Delta t}\hat\mathbf y + \frac{\Delta z}{\Delta t}\hat\mathbf z =\bar v_x\hat\mathbf x + \bar v_y\hat\mathbf y + \bar v_z \hat\mathbf z \,</math>where <math>\Delta \mathbf{r}</math> is the displacement vector during the time interval <math>\Delta t</math>. In the limit that the time interval <math>\Delta t</math> approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector,
<math display="block"> \mathbf v
= \lim_{\Delta t\to 0}\frac{\Delta\mathbf{r}}{\Delta t}
= \frac{\text{d}\mathbf r}{\text{d}t}
= v_x\hat\mathbf x + v_y\hat\mathbf y + v_z \hat\mathbf z .</math>
Thus, a particle's velocity is the time rate of change of its position. Furthermore, this velocity is [[tangent]] to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.

The [[speed]] of an object is the magnitude of its velocity. It is a scalar quantity:
<math display="block"> v=|\mathbf{v}|= \frac {\text{d}s}{\text{d}t},</math>
where <math>s</math> is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, <math>\frac{\text{d}s}{\text{d}t}</math> is non-negative, which implies that speed is also non-negative.

===Acceleration===
The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both the rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The [[acceleration]] of a particle is the vector defined by the rate of change of the velocity vector. The [[average acceleration]] of a particle over a time interval is defined as the ratio.
<math display="block"> \mathbf\bar a = \frac{\Delta \mathbf\bar v}{\Delta t} = \frac{\Delta \bar v_x}{\Delta t}\hat\mathbf x + \frac{\Delta \bar v_y}{\Delta t}\hat\mathbf y + \frac{\Delta \bar v_z}{\Delta t}\hat\mathbf z =\bar a_x\hat\mathbf x + \bar a_y\hat\mathbf y + \bar a_z \hat\mathbf z \,</math>
where Δ'''v''' is the average velocity and Δ''t'' is the time interval.

The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative,
<math display="block"> \mathbf a
= \lim_{\Delta t\to 0}\frac{\Delta\mathbf{v}}{\Delta t}
=\frac{\text{d}\mathbf v}{\text{d}t}
= a_x\hat\mathbf x + a_y\hat\mathbf y + a_z \hat\mathbf z  . </math>

Alternatively,
<math display="block"> \mathbf a
= \lim_{(\Delta t)^2 \to 0}\frac{\Delta\mathbf{r}}{(\Delta t)^2}
= \frac{\text{d}^2\mathbf r}{\text{d}t^2}
= a_x\hat\mathbf x + a_y\hat\mathbf y + a_z \hat\mathbf z . </math>

Thus, acceleration is the first derivative of the velocity vector and the second derivative of the position vector of that particle. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.

The magnitude of the [[acceleration]] of an object is the magnitude |'''a'''| of its acceleration vector. It is a scalar quantity:
<math display="block"> |\mathbf{a}| = |\dot{\mathbf{v}} | = \frac{\text{d}v}{\text{d}t}.</math>

===Relative position vector===
[[Displacement (vector)|A relative position vector]] is a vector that defines the position of one point relative to another. It is the difference in position of the two points.
The position of one point ''A'' relative to another point ''B'' is simply the difference between their positions

:<math>\mathbf{r}_{A/B} = \mathbf{r}_{A} - \mathbf{r}_{B} </math>

which is the difference between the components of their position vectors.

If point ''A'' has position components <math>\mathbf{r}_{A} = \left( x_{A}, y_{A}, z_{A} \right) </math>

and point ''B'' has position components <math>\mathbf{r}_{B} = \left( x_{B}, y_{B}, z_{B} \right) </math>

then the position of point ''A'' relative to point ''B'' is the difference between their components: <math>\mathbf{r}_{A/B} = \mathbf{r}_{A} - \mathbf{r}_{B} = \left( x_{A} - x_{B}, y_{A} - y_{B}, z_{A} - z_{B} \right) </math>

===Relative velocity===
{{main|Relative velocity}}
[[File:Relative velocity.svg|300px|thumb|Relative velocities between two particles in classical mechanics.]]

The velocity of one point relative to another is simply the difference between their velocities
<math display="block">\mathbf{v}_{A/B} = \mathbf{v}_{A} - \mathbf{v}_{B} </math>
which is the difference between the components of their velocities.

If point ''A'' has velocity components <math>\mathbf{v}_{A} = \left( v_{A_x}, v_{A_y}, v_{A_z} \right) </math> and point ''B'' has velocity components <math>\mathbf{v}_{B} = \left( v_{B_x}, v_{B_y}, v_{B_z} \right) </math> then the velocity of point ''A'' relative to point ''B'' is the difference between their components:
<math>\mathbf{v}_{A/B} = \mathbf{v}_{A} - \mathbf{v}_{B} = \left( v_{A_x} - v_{B_x}, v_{A_y} - v_{B_{y}}, v_{A_z} - v_{B_z} \right) </math>

Alternatively, this same result could be obtained by computing the time derivative of the relative position vector '''r'''<sub>B/A</sub>.

===Relative acceleration===

The acceleration of one point ''C'' relative to another point ''B'' is simply the difference between their accelerations.
<math display="block">\mathbf{a}_{C/B} = \mathbf{a}_{C} - \mathbf{a}_{B} </math>
which is the difference between the components of their accelerations.

If point ''C'' has acceleration components <math>\mathbf{a}_{C} = \left( a_{C_x}, a_{C_y}, a_{C_z} \right) </math>
and point ''B'' has acceleration components <math>\mathbf{a}_{B} = \left( a_{B_x}, a_{B_y}, a_{B_z} \right) </math>
then the acceleration of point ''C'' relative to point ''B'' is the difference between their components: <math>\mathbf{a}_{C/B} = \mathbf{a}_{C} - \mathbf{a}_{B} = \left( a_{C_x} - a_{B_x} , a_{C_y} - a_{B_y} , a_{C_z} - a_{B_z} \right) </math>

Assuming that the initial conditions of the position, <math>\mathbf{r}_0</math>, and velocity <math>\mathbf{v}_0</math> at time <math>t = 0</math> are known, the first integration yields the velocity of the particle as a function of time.<ref>{{Citation | title=2.4 Integration | date=2 June 2017 | url=https://www.youtube.com/watch?v=H7xmTMQ265s | archive-url=https://ghostarchive.org/varchive/youtube/20211113/H7xmTMQ265s| archive-date=2021-11-13 | url-status=live| publisher=MIT | language=en | access-date=2021-07-04}}{{cbignore}}</ref>
<math display="block">\mathbf{v}(t) = \mathbf{v}_0 + \int_0^t \mathbf{a}(\tau) \, \text{d}\tau</math>

Additional relations between displacement, velocity, acceleration, and time can be derived. If the acceleration is constant,
<math display="block">\mathbf{a} = \frac{\Delta\mathbf{v}}{\Delta t} = \frac{\mathbf{v}-\mathbf{v}_0}{ t } </math> can be substituted into the above equation to give:
<math display="block">\mathbf{r}(t) = \mathbf{r}_0 + \left(\frac{\mathbf{v} + \mathbf{v}_0}{2}\right) t .</math>

A relationship between velocity, position and acceleration without explicit time dependence can be obtained by solving the average acceleration for time and substituting and simplifying

<math display="block"> t = \frac{\mathbf{v}-\mathbf{v}_0}{ \mathbf{a} } </math>

<math display="block"> \left(\mathbf{r} - \mathbf{r}_0\right) \cdot \mathbf{a} = \left( \mathbf{v} - \mathbf{v}_0 \right) \cdot \frac{\mathbf{v} + \mathbf{v}_0}{2} \ , </math>
where <math> \cdot </math> denotes the [[dot product]], which is appropriate as the products are scalars rather than vectors.
<math display="block">2 \left(\mathbf{r} - \mathbf{r}_0\right) \cdot \mathbf{a} = |\mathbf{v}|^2 - |\mathbf{v}_0|^2.</math>

The dot product can be replaced by the cosine of the angle {{mvar|α}} between the vectors (see [[Dot product#Geometric definition|Geometric interpretation of the dot product]] for more details) and the vectors by their magnitudes, in which case:
<math display="block">2 \left|\mathbf{r} - \mathbf{r}_0\right| \left|\mathbf{a}\right| \cos \alpha = |\mathbf{v}|^2 - |\mathbf{v}_0|^2.</math>

In the case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors ({{mvar|α}}) is 0, so <math>\cos 0 = 1</math>, and
<math display="block"> |\mathbf{v}|^2= |\mathbf{v}_0|^2 + 2 \left|\mathbf{a}\right| \left|\mathbf{r}-\mathbf{r}_0\right|.</math>
This can be simplified using the notation for the magnitudes of the vectors <math>|\mathbf{a}|=a, |\mathbf{v}|=v, |\mathbf{r}-\mathbf{r}_0| = \Delta r </math>{{citation needed|date=April 2018}} where <math>\Delta r</math> can be any curvaceous path taken as the constant tangential acceleration is applied along that path{{citation needed|date=April 2018}}, so
<math display="block"> v^2= v_0^2 + 2a \Delta r.</math>
This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that <math display="inline">\Delta r = \int v \, \text{d}t</math> or <math>\Delta r</math> is the area under a velocity–time graph.<ref>https://www.youtube.com/watch?v=jLJLXka2wEM Crash course physics integrals</ref> [[File:Velocity Time physics graph.svg|thumb|Velocity Time physics graph]] We can take <math>\Delta r</math> by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the <math>A \cdot B</math> where <math>A</math> is the width and <math>B</math> is the height. In this case <math>A = t</math> and <math>B = v_0</math> (the <math>A</math> here is different from the acceleration <math>a</math>). This means that the bottom area is <math>tv_0</math>. Now let's find the top area (a triangle). The area of a triangle is <math display="inline">\frac{1}{2} BH</math> where <math>B</math> is the base and <math>H</math> is the height.<ref>https://www.mathsisfun.com/algebra/trig-area-triangle-without-right-angle.html Area of Triangles Without Right Angles</ref> In this case, <math>B = t</math> and <math>H = at</math> or <math display="inline">A = \frac{1}{2} BH = \frac{1}{2} att = \frac{1}{2} at^2 = \frac{at^2}{2}</math>. Adding <math>v_0 t</math> and <math display="inline">\frac{at^2}{2}</math> results in the equation <math>\Delta r</math> results in the equation <math display="inline">\Delta r = v_0 t + \frac{at^2}{2}</math>.<ref>{{Cite AV media |url=https://www4.uwsp.edu/physastr/kmenning/Phys203/eqs/kinematics.gif |title=kinematics.gif (508×368) |type=Image |language=en |access-date=3 November 2023}}</ref> This equation is applicable when the final velocity {{mvar|v}} is unknown.

[[File:Nonuniform circular motion.svg|thumb|250px|Figure 2: Velocity and acceleration for nonuniform circular motion: the velocity vector is tangential to the orbit, but the acceleration vector is not radially inward because of its tangential component '''a'''<sub>''θ''</sub> that increases the rate of rotation: d''ω''/d''t'' = <nowiki>|</nowiki>'''a'''<sub>''θ''</sub><nowiki>|</nowiki>/''R''.]]

== Particle trajectories in cylindrical-polar coordinates ==
{{see also|Generalized coordinates|Curvilinear coordinates|Orthogonal coordinates|Frenet-Serret formulas}}

It is often convenient to formulate the trajectory of a particle '''r'''(''t'') = (''x''(''t''), ''y''(''t''), ''z''(''t'')) using polar coordinates in the ''X''–''Y'' plane. In this case, its velocity and acceleration take a convenient form.

Recall that the trajectory of a particle ''P'' is defined by its coordinate vector '''r''' measured in a fixed reference frame ''F''. As the particle moves, its coordinate vector '''r'''(''t'') traces its trajectory, which is a curve in space, given by:
<math display="block"> \mathbf r(t) = x(t)\hat\mathbf x + y(t) \hat\mathbf y +z(t) \hat\mathbf z,</math>
where '''x̂''', '''ŷ''', and '''ẑ''' are the [[unit vectors]] along the ''x'', ''y'' and ''z'' axes of the [[Cartesian coordinate system|reference frame]] ''F'', respectively.

Consider a particle ''P'' that moves only on the surface of a circular cylinder ''r''(''t'') = constant, it is possible to align the ''z'' axis of the fixed frame ''F'' with the axis of the cylinder. Then, the angle ''θ'' around this axis in the ''x''–''y'' plane can be used to define the trajectory as,
<math display="block"> \mathbf{r}(t) = r\cos(\theta(t))\hat\mathbf x + r\sin(\theta(t))\hat\mathbf y + z(t)\hat\mathbf z, </math>
where the constant distance from the center is denoted as ''r'', and ''θ''(''t'') is a function of time.

The cylindrical coordinates for '''r'''(''t'') can be simplified by introducing the radial and tangential unit vectors,
<math display="block"> \hat\mathbf r = \cos(\theta(t))\hat\mathbf x + \sin(\theta(t))\hat\mathbf y,
\quad
\hat\mathbf\theta = -\sin(\theta(t))\hat\mathbf x + \cos(\theta(t))\hat\mathbf y .</math>
and their time derivatives from elementary calculus:
<math display="block"> \frac{\text{d}\hat\mathbf r}{\text{d}t} = \omega\hat\mathbf\theta . </math>
<math display="block"> \frac{\text{d}^2\hat\mathbf r}{\text{d}t^2} = \frac{\text{d}(\omega\hat\mathbf\theta)}{\text{d}t}  = \alpha\hat\mathbf\theta - \omega^2\hat\mathbf r  . </math>

<math display="block"> \frac{\text{d}\hat\mathbf\theta}{\text{d}t} = -\omega\hat\mathbf r . </math>
<math display="block"> \frac{\text{d}^2\hat\mathbf\theta}{\text{d}t^2} = \frac{\text{d}(-\omega\hat\mathbf r)}{\text{d}t} = -\alpha\hat\mathbf r - \omega^2\hat\mathbf\theta. </math>

Using this notation, '''r'''(''t'') takes the form,
<math display="block"> \mathbf{r}(t) = r\hat\mathbf r + z(t)\hat\mathbf z .</math>
In general, the trajectory '''r'''(''t'') is not constrained to lie on a circular cylinder, so the radius ''R'' varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes:
<math display="block"> \mathbf{r}(t) = r(t)\hat\mathbf r + z(t)\hat\mathbf z .</math>
Where ''r'', ''θ'', and ''z'' might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector '''v'''<sub>''P''</sub> is the time derivative of the trajectory '''r'''(''t''), which yields:
<math display="block"> \mathbf v_P
= \frac{\text{d}}{\text{d}t} \left(r\hat\mathbf r + z \hat\mathbf z \right)
= v\hat\mathbf r + r\mathbf\omega\hat\mathbf\theta + v_z\hat\mathbf z = v(\hat\mathbf r + \hat\mathbf\theta) + v_z\hat\mathbf z  . </math>

Similarly, the acceleration '''a'''<sub>''P''</sub>, which is the time derivative of the velocity '''v'''<sub>''P''</sub>, is given by:
<math display="block"> \mathbf{a}_P
= \frac{\text{d}}{\text{d}t} \left(v\hat\mathbf r + v\hat\mathbf\theta + v_z\hat\mathbf z\right) =(a - v\omega)\hat\mathbf r + (a + v\omega)\hat\mathbf\theta + a_z\hat\mathbf z . </math>

The term <math> -v\omega\hat\mathbf r</math> acts toward the center of curvature of the path at that point on the path, is commonly called the [[centripetal acceleration]]. The term <math> v\omega\hat\mathbf\theta</math> is called the [[Coriolis acceleration]].

===Constant radius===
If the trajectory of the particle is constrained to lie on a cylinder, then the radius ''r'' is constant and the velocity and acceleration vectors simplify. The velocity of '''v'''<sub>P</sub> is the time derivative of the trajectory '''r'''(''t''),
<math display="block"> \mathbf v_P
= \frac{\text{d}}{\text{d}t} \left(r\hat\mathbf r + z \hat\mathbf z \right)
= r\omega\hat\mathbf\theta + v_z\hat\mathbf z = v\hat\mathbf\theta + v_z\hat\mathbf z .</math>

===Planar circular trajectories===
[[File:The Kinematics of Machinery - Figure 3.jpg|thumb|right|300px|alt=Kinematics of Machinery|Each particle on the wheel travels in a planar circular trajectory (Kinematics of Machinery, 1876).<ref>{{citation |last1=Reuleaux |first1=F. |author-link=Franz Reuleaux |year=1876 |title=The Kinematics of Machinery: Outlines of a Theory of Machines |first2=Alex B. W. |last2=Kennedy |publisher=Macmillan |location=London |url=https://archive.org/details/kinematicsofmach00reuluoft}}</ref>]]
A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the ''z'' axis:
<math display="block"> \mathbf r(t) = r\hat\mathbf r + z \hat\mathbf z,</math>
where ''r'' and ''z''<sub>0</sub> are constants. In this case, the velocity '''v'''<sub>''P''</sub> is given by:
<math display="block"> \mathbf{v}_P
= \frac{\text{d}}{\text{d}t} \left(r\hat\mathbf r + z \hat\mathbf z\right)
= r\omega\hat\mathbf\theta = v\hat\mathbf\theta,</math>
where <math> \omega</math> is the [[angular velocity]] of the unit vector {{math|θ<sup>^</sup>}} around the ''z'' axis of the cylinder.

The acceleration '''a'''<sub>''P''</sub> of the particle ''P'' is now given by:
<math display="block"> \mathbf{a}_P = \frac{\text{d}(v\hat\mathbf\theta)}{\text{d}t} = a\hat\mathbf\theta - v\theta\hat\mathbf r.</math>

The components
<math display="block"> a_r = - v\theta, \quad a_{\theta} = a,</math>
are called, respectively, the ''radial'' and ''tangential components'' of acceleration.

The notation for angular velocity and [[angular acceleration]] is often defined as
<math display="block">\omega = \dot{\theta}, \quad \alpha = \ddot{\theta}, </math>
so the radial and tangential acceleration components for circular trajectories are also written as
<math display="block"> a_r = - r\omega^2, \quad a_{\theta} = r\alpha.</math>

== Point trajectories in a body moving in the plane ==

The movement of components of a [[mechanical system]] are analyzed by attaching a [[Cartesian coordinate system|reference frame]] to each part and determining how the various reference frames move relative to each other. If the structural stiffness of the parts are sufficient, then their deformation can be neglected and rigid transformations can be used to define this relative movement. This reduces the description of the motion of the various parts of a complicated mechanical system to a problem of describing the geometry of each part and geometric association of each part relative to other parts.

[[Geometry]] is the study of the properties of figures that remain the same while the space is transformed in various ways—more technically, it is the study of invariants under a set of transformations.<ref>Geometry: the study of properties of given elements that remain invariant under specified transformations. {{cite web |title=Definition of geometry |date=31 May 2023 |publisher=Merriam-Webster on-line dictionary |url=http://www.merriam-webster.com/dictionary/geometry}}</ref> These transformations can cause the displacement of the triangle in the plane, while leaving the vertex angle and the distances between vertices unchanged. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry.

The coordinates of points in a plane are two-dimensional vectors in '''R'''<sup>2</sup> (two dimensional space). Rigid transformations are those that preserve the [[distance formula|distance]] between any two points. The set of rigid transformations in an ''n''-dimensional space is called the special [[Euclidean group]] on '''R'''<sup>''n''</sup>, and denoted [[SE(n)|SE(''n'')]].

===Displacements and motion===
[[File:SteamEngine Boulton&Watt 1784.png|thumb|right|300px|alt=Boulton & Watt Steam Engine|The movement of each of the components of the Boulton & Watt Steam Engine (1784) is modeled by a continuous set of rigid displacements.]]
The position of one component of a mechanical system relative to another is defined by introducing a reference frame, say ''M'', on one that moves relative to a fixed frame, ''F,'' on the other. The rigid transformation, or displacement, of ''M'' relative to ''F'' defines the relative position of the two components. A displacement consists of the combination of a [[rotation]] and a [[translation (geometry)|translation]].

The set of all displacements of ''M'' relative to ''F'' is called the [[Configuration space (physics)|configuration space]] of ''M.'' A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the [[motion (physics)|motion]] of ''M'' relative to ''F.'' The motion of a body consists of a continuous set of rotations and translations.

===Matrix representation===
The combination of a rotation and translation in the plane '''R'''<sup>2</sup> can be represented by a certain type of 3×3 matrix known as a homogeneous transform. The 3×3 homogeneous transform is constructed from a 2×2 [[rotation matrix]] ''A''(''φ'') and the 2×1 translation vector '''d''' = (''d<sub>x</sub>'', ''d<sub>y</sub>''), as:
<math display="block"> [T(\phi, \mathbf{d})]
= \begin{bmatrix} A(\phi) & \mathbf{d} \\ \mathbf 0 & 1\end{bmatrix}
= \begin{bmatrix} \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end{bmatrix}.</math>
These homogeneous transforms perform rigid transformations on the points in the plane ''z'' = 1, that is, on points with coordinates '''r''' = (''x'', ''y'', 1).

In particular, let '''r''' define the coordinates of points in a reference frame ''M'' coincident with a fixed frame ''F''. Then, when the origin of ''M'' is displaced by the translation vector '''d''' relative to the origin of ''F'' and rotated by the angle φ relative to the x-axis of ''F'', the new coordinates in ''F'' of points in ''M'' are given by:
<math display="block"> \mathbf{P} = [T(\phi, \mathbf{d})]\mathbf{r}
= \begin{bmatrix} \cos\phi & -\sin\phi & d_x \\ \sin\phi & \cos\phi & d_y \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix}x\\y\\1\end{bmatrix}.</math>

Homogeneous transforms represent [[affine transformation]]s. This formulation is necessary because a [[translation (geometry)|translation]] is not a [[linear transformation]] of '''R'''<sup>2</sup>. However, using projective geometry, so that '''R'''<sup>2</sup> is considered a subset of '''R'''<sup>3</sup>, translations become affine linear transformations.<ref>{{cite book |last=Paul |first=Richard |title=Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators |publisher=MIT Press, Cambridge, MA |year=1981 |url=https://books.google.com/books?id=UzZ3LAYqvRkC|isbn=978-0-262-16082-7}}</ref>

== Pure translation ==

If a rigid body moves so that its [[Cartesian coordinate system|reference frame]] ''M'' does not rotate (''θ'' = 0) relative to the fixed frame ''F'', the motion is called pure translation. In this case, the trajectory of every point in the body is an offset of the trajectory '''d'''(''t'') of the origin of ''M,'' that is:
<math display="block"> \mathbf{r}(t)=[T(0,\mathbf{d}(t))] \mathbf{p} = \mathbf{d}(t) + \mathbf{p}.</math>

Thus, for bodies in pure translation, the velocity and acceleration of every point ''P'' in the body are given by:
<math display="block"> \mathbf{v}_P=\dot{\mathbf{r}}(t) = \dot{\mathbf{d}}(t)=\mathbf{v}_O,
\quad
\mathbf{a}_P=\ddot{\mathbf{r}}(t) = \ddot{\mathbf{d}}(t) = \mathbf{a}_O,</math>
where the dot denotes the derivative with respect to time and '''v'''<sub>''O''</sub> and '''a'''<sub>''O''</sub> are the velocity and acceleration, respectively, of the origin of the moving frame ''M''. Recall the coordinate vector '''p''' in ''M'' is constant, so its derivative is zero.

== Rotation of a body around a fixed axis ==
{{main|Rotation around a fixed axis}}
[[File:Rotating body.PNG|thumb|250px|Figure 1: The angular velocity vector '''Ω''' points up for counterclockwise rotation and down for clockwise rotation, as specified by the [[right-hand rule]]. Angular position ''θ''(''t'') changes with time at a rate {{nowrap|1=''ω''(''t'') = d''θ''/d''t''}}.]]

Objects like a playground [[merry-go-round]], ventilation fans, or hinged doors can be modeled as rigid bodies rotating about a single fixed axis.<ref name=Gregory-2006>{{Cite book |last=Gregory |first=R. Douglas |url=https://www.cambridge.org/core/product/identifier/9780511803789/type/book |title=Classical Mechanics |date=2006-04-13 |publisher=Cambridge University Press |isbn=978-0-521-82678-5 |edition=1 |doi=10.1017/cbo9780511803789}}</ref>{{rp|37}} The ''z''-axis has been chosen by convention.

=== Position ===
This allows the description of a rotation as the angular position of a planar reference frame ''M'' relative to a fixed ''F'' about this shared ''z''-axis. Coordinates '''p''' = (''x'', ''y'') in ''M'' are related to coordinates '''P''' = (X, Y) in ''F'' by the matrix equation:
<math display="block"> \mathbf{P}(t) = [A(t)]\mathbf{p}, </math>

where
<math display="block"> [A(t)] = \begin{bmatrix}
 \cos(\theta(t)) & -\sin(\theta(t)) \\
 \sin(\theta(t)) & \cos(\theta(t))
\end{bmatrix}, </math>
is the rotation matrix that defines the angular position of ''M'' relative to ''F'' as a function of time.

=== Velocity ===
If the point '''p''' does not move in ''M'', its velocity in ''F'' is given by
<math display="block"> \mathbf{v}_P = \dot{\mathbf{P}} = [\dot{A}(t)]\mathbf{p}. </math>
It is convenient to eliminate the coordinates '''p''' and write this as an operation on the trajectory '''P'''(''t''),
<math display="block"> \mathbf{v}_P = [\dot{A}(t)][A(t)^{-1}]\mathbf{P} = [\Omega]\mathbf{P}, </math>
where the matrix
<math display="block"> [\Omega] = \begin{bmatrix} 0 & -\omega \\ \omega & 0 \end{bmatrix}, </math>
is known as the angular velocity matrix of ''M'' relative to ''F''. The parameter ''ω'' is the time derivative of the angle ''θ'', that is:
<math display="block"> \omega = \frac{\text{d}\theta}{\text{d}t}. </math>

=== Acceleration ===
The acceleration of '''P'''(''t'') in ''F'' is obtained as the time derivative of the velocity,
<math display="block"> \mathbf{A}_P = \ddot{P}(t) = [\dot{\Omega}]\mathbf{P} + [\Omega]\dot{\mathbf{P}}, </math>
which becomes
<math display="block"> \mathbf{A}_P = [\dot{\Omega}]\mathbf{P} + [\Omega][\Omega]\mathbf{P}, </math>
where
<math display="block"> [\dot{\Omega}] = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix}, </math>
is the angular acceleration matrix of ''M'' on ''F'', and
<math display="block"> \alpha = \frac{\text{d}^2\theta}{\text{d}t^2}. </math>

The description of rotation then involves these three quantities:

* '''Angular position''': the oriented distance from a selected origin on the rotational axis to a point of an object is a vector '''r'''(''t'') locating the point. The vector '''r'''(''t'') has some projection (or, equivalently, some component) '''r'''<sub>⊥</sub>(''t'') on a plane perpendicular to the axis of rotation. Then the ''angular position'' of that point is the angle ''θ'' from a reference axis (typically the positive ''x''-axis) to the vector '''r'''<sub>⊥</sub>(''t'') in a known rotation sense (typically given by the [[right-hand rule]]).
* '''Angular velocity''': the angular velocity ''ω'' is the rate at which the angular position ''θ'' changes with respect to time ''t'': <math display="block">\omega = \frac {\text{d}\theta}{\text{d}t}</math> The angular velocity is represented in Figure 1 by a vector '''Ω''' pointing along the axis of rotation with magnitude ''ω'' and sense determined by the direction of rotation as given by the [[right-hand rule]].
* '''Angular acceleration''': the magnitude of the angular acceleration ''α'' is the rate at which the angular velocity ''ω'' changes with respect to time ''t'': <math display="block">\alpha = \frac {\text{d}\omega}{\text{d}t}</math>

The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges:
<math display="block">\omega_{\mathrm{f}} = \omega_{\mathrm{i}} + \alpha t\!</math>
<math display="block">\theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \omega_{\mathrm{i}} t + \tfrac{1}{2} \alpha t^2</math>
<math display="block">\theta_{\mathrm{f}} - \theta_{\mathrm{i}} = \tfrac{1}{2} (\omega_{\mathrm{f}} + \omega_{\mathrm{i}})t</math>
<math display="block">\omega_{\mathrm{f}}^2 = \omega_{\mathrm{i}}^2 + 2 \alpha (\theta_{\mathrm{f}} - \theta_{\mathrm{i}}).</math>

Here ''θ''<sub>i</sub> and ''θ''<sub>f</sub> are, respectively, the initial and final angular positions, ''ω''<sub>i</sub> and ''ω''<sub>f</sub> are, respectively, the initial and final angular velocities, and ''α'' is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector.

== Point trajectories in body moving in three dimensions ==

Important formulas in kinematics define the [[velocity]] and acceleration of points in a moving body as they trace trajectories in three-dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either [[Newton's second law]] or [[Lagrangian mechanics|Lagrange's equations]].

===Position===
In order to define these formulas, the movement of a component ''B'' of a mechanical system is defined by the set of rotations [A(''t'')] and translations '''d'''(''t'') assembled into the homogeneous transformation [T(''t'')]=[A(''t''), '''d'''(''t'')]. If '''p''' is the coordinates of a point ''P'' in ''B'' measured in the moving [[Cartesian coordinate system|reference frame]] ''M'', then the trajectory of this point traced in ''F'' is given by:
<math display="block"> \mathbf{P}(t) = [T(t)] \mathbf{p}
= \begin{bmatrix} \mathbf{P} \\ 1\end{bmatrix}
=\begin{bmatrix} A(t) & \mathbf{d}(t) \\ 0 & 1\end{bmatrix}
\begin{bmatrix} \mathbf{p} \\ 1\end{bmatrix}.</math>
This notation does not distinguish between '''P''' = (X, Y, Z, 1), and '''P''' = (X, Y, Z), which is hopefully clear in context.

This equation for the trajectory of ''P'' can be inverted to compute the coordinate vector '''p''' in ''M'' as:
<math display="block"> \mathbf{p} = [T(t)]^{-1}\mathbf{P}(t)
= \begin{bmatrix} \mathbf{p} \\ 1\end{bmatrix}
=\begin{bmatrix} A(t)^\text{T} & -A(t)^\text{T}\mathbf{d}(t) \\ 0 & 1\end{bmatrix}
\begin{bmatrix} \mathbf{P}(t) \\ 1\end{bmatrix}.</math>
This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is:
<math display="block"> [A(t)]^\text{T}[A(t)]=I.\!</math>

===Velocity===
The velocity of the point ''P'' along its trajectory '''P'''(''t'') is obtained as the time derivative of this position vector,
<math display="block"> \mathbf{v}_P = [\dot{T}(t)]\mathbf{p}
=\begin{bmatrix} \mathbf{v}_P \\ 0\end{bmatrix}
= \left(\frac{d}{dt}{\begin{bmatrix} A(t) & \mathbf{d}(t) \\ 0 & 1 \end{bmatrix}}\right)
\begin{bmatrix} \mathbf{p} \\ 1\end{bmatrix}
= \begin{bmatrix} \dot{A}(t) & \dot{\mathbf{d}}(t) \\ 0 & 0 \end{bmatrix}
\begin{bmatrix} \mathbf{p} \\ 1\end{bmatrix}.</math>
The dot denotes the derivative with respect to time; because '''p''' is constant, its derivative is zero.

This formula can be modified to obtain the velocity of ''P'' by operating on its trajectory '''P'''(''t'') measured in the fixed frame ''F''. Substituting the [[Inverse Laplace transform|inverse transform]] for '''p''' into the velocity equation yields:
<math display="block">\begin{align}
\mathbf{v}_P & = [\dot{T}(t)][T(t)]^{-1}\mathbf{P}(t) \\[4pt]
& =
\begin{bmatrix} \mathbf{v}_P \\ 0 \end{bmatrix}
 =
\begin{bmatrix} \dot{A} & \dot{\mathbf{d}} \\ 0 & 0 \end{bmatrix}
\begin{bmatrix} A & \mathbf{d} \\ 0 & 1 \end{bmatrix}^{-1}
\begin{bmatrix} \mathbf{P}(t) \\ 1\end{bmatrix} \\[4pt]
& =
\begin{bmatrix} \dot{A} & \dot{\mathbf{d}} \\ 0 & 0 \end{bmatrix}
A^{-1}\begin{bmatrix} 1 & -\mathbf{d} \\ 0 & A \end{bmatrix}
\begin{bmatrix} \mathbf{P}(t) \\ 1\end{bmatrix} \\[4pt]
& =
\begin{bmatrix} \dot{A}A^{-1} & -\dot{A}A^{-1}\mathbf{d} + \dot{\mathbf{d}} \\ 0 & 0 \end{bmatrix}
\begin{bmatrix} \mathbf{P}(t) \\ 1\end{bmatrix} \\[4pt]
&=
\begin{bmatrix} \dot{A}A^\text{T} & -\dot{A}A^\text{T}\mathbf{d} + \dot{\mathbf{d}} \\ 0 & 0 \end{bmatrix}
\begin{bmatrix} \mathbf{P}(t) \\ 1\end{bmatrix} \\[6pt]
\mathbf{v}_P &= [S]\mathbf{P}.
\end{align}</math>
The matrix [''S''] is given by:
<math display="block"> [S] = \begin{bmatrix} \Omega & -\Omega\mathbf{d} + \dot{\mathbf{d}} \\ 0 & 0 \end{bmatrix}</math>
where
<math display="block"> [\Omega] = \dot{A}A^\text{T},</math>
is the angular velocity matrix.

Multiplying by the operator [''S''], the formula for the velocity '''v'''<sub>P</sub> takes the form:
<math display="block">\mathbf{v}_P = [\Omega](\mathbf{P}-\mathbf{d}) + \dot{\mathbf{d}} = \omega\times \mathbf{R}_{P/O} + \mathbf{v}_O,</math>
where the vector ''ω'' is the angular velocity vector obtained from the components of the matrix [Ω]; the vector
<math display="block"> \mathbf{R}_{P/O}=\mathbf{P}-\mathbf{d},</math>
is the position of ''P'' relative to the origin ''O'' of the moving frame ''M''; and
<math display="block">\mathbf{v}_O=\dot{\mathbf{d}},</math>
is the velocity of the origin ''O''.

===Acceleration===
The acceleration of a point ''P'' in a moving body ''B'' is obtained as the time derivative of its velocity vector:
<math display="block">\mathbf{A}_P = \frac{d}{dt}\mathbf{v}_P
= \frac{d}{dt}\left([S]\mathbf{P}\right)
= [\dot{S}] \mathbf{P} + [S] \dot{\mathbf{P}}
= [\dot{S}]\mathbf{P} + [S][S]\mathbf{P} .</math>

This equation can be expanded firstly by computing
<math display="block"> [\dot{S}] = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\mathbf{d} -\Omega\dot{\mathbf{d}} + \ddot{\mathbf{d}} \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} \dot{\Omega} & -\dot{\Omega}\mathbf{d} -\Omega\mathbf{v}_O + \mathbf{A}_O \\ 0 & 0 \end{bmatrix}</math>
and
<math display="block"> [S]^2 = \begin{bmatrix} \Omega & -\Omega\mathbf{d} + \mathbf{v}_O \\ 0 & 0 \end{bmatrix}^2 = \begin{bmatrix} \Omega^2 & -\Omega^2\mathbf{d} + \Omega\mathbf{v}_O \\ 0 & 0 \end{bmatrix}.</math>

The formula for the acceleration '''A'''<sub>''P''</sub> can now be obtained as:
<math display="block"> \mathbf{A}_P = \dot{\Omega}(\mathbf{P} - \mathbf{d}) + \mathbf{A}_O + \Omega^2(\mathbf{P}-\mathbf{d}),</math>
or
<math display="block"> \mathbf{A}_P = \alpha\times\mathbf{R}_{P/O} + \omega\times\omega\times\mathbf{R}_{P/O} + \mathbf{A}_O,</math>
where ''α'' is the angular acceleration vector obtained from the derivative of the angular velocity vector;
<math display="block">\mathbf{R}_{P/O}=\mathbf{P}-\mathbf{d},</math>
is the relative position vector (the position of ''P'' relative to the origin ''O'' of the moving frame ''M''); and
<math display="block">\mathbf{A}_O = \ddot{\mathbf{d}}</math>
is the acceleration of the origin of the moving frame ''M''.

== Kinematic constraints ==

Kinematic constraints are constraints on the movement of components of a mechanical system. Kinematic constraints can be considered to have two basic forms, (i) constraints that arise from hinges, sliders and cam joints that define the construction of the system, called [[holonomic system|holonomic constraints]], and (ii) constraints imposed on the velocity of the system such as the knife-edge constraint of ice-skates on a flat plane, or rolling without slipping of a disc or sphere in contact with a plane, which are called [[non-holonomic system|non-holonomic constraints]]. The following are some common examples.

===Kinematic coupling===
A [[kinematic coupling]] exactly constrains all 6 degrees of freedom.

===Rolling without slipping===
An object that rolls against a [[Surface (topology)|surface]] without slipping obeys the condition that the [[velocity]] of its [[center of mass]] is equal to the [[cross product]] of its angular velocity with a vector from the point of contact to the center of mass:
<math display="block"> \boldsymbol{ v}_G(t) = \boldsymbol{\Omega} \times \boldsymbol{ r}_{G/O}.</math>

For the case of an object that does not tip or turn, this reduces to <math> v = r \omega</math>.

===Inextensible cord===
This is the case where bodies are connected by an idealized cord that remains in tension and cannot change length. The constraint is that the sum of lengths of all segments of the cord is the total length, and accordingly the time derivative of this sum is zero.<ref name="Kelvin">{{cite book |title=Elements of Natural Philosophy |page=[https://archive.org/details/elementsnatural00kelvgoog/page/n16 4] |author=William Thomson Kelvin & Peter Guthrie Tait |url=https://archive.org/details/elementsnatural00kelvgoog|year=1894 |publisher=Cambridge University Press |isbn=1-57392-984-0}}</ref><ref name="Thompson2">{{cite book |title=Elements of Natural Philosophy |page=296 |author=William Thomson Kelvin & Peter Guthrie Tait |url=https://books.google.com/books?id=ahtWAAAAMAAJ&pg=PA296|year=1894}}</ref><ref name="Fogiel">{{cite book |title=The Mechanics Problem Solver |author=M. Fogiel |year=1980 |isbn=0-87891-519-2 |publisher=Research & Education Association |chapter-url=https://books.google.com/books?id=XVyD9pJpW-cC&q=%22inextensible+cord%22&pg=PA613 |chapter=Problem 17-11 |page=613}}</ref> A dynamic problem of this type is the [[pendulum]]. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord.<ref name="Church">
{{cite book |title=Mechanics of Engineering |author=Irving Porter Church |url=https://archive.org/details/mechanicsengine05churgoog|page=[https://archive.org/details/mechanicsengine05churgoog/page/n139 111] |publisher=Wiley |year=1908 |isbn=1-110-36527-6}}</ref> An ''equilibrium'' problem (i.e. not kinematic) of this type is the [[catenary]].<ref name="Kline">{{cite book |title=Mathematical Thought from Ancient to Modern Times |author= Morris Kline |url=https://archive.org/details/mathematicalthou00klin|url-access=registration |page= [https://archive.org/details/mathematicalthou00klin/page/472 472] |isbn=0-19-506136-5 |publisher=Oxford University Press |year=1990}}</ref>

===Kinematic pairs===
{{main|Kinematic pair}}
[[Franz Reuleaux|Reuleaux]] called the ideal connections between components that form a machine [[kinematic pair]]s. He distinguished between higher pairs which were said to have line contact between the two links and lower pairs that have area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple classification.<ref>{{cite book |last=Phillips |first=Jack |title=Freedom in Machinery, Volumes 1–2 |publisher=Cambridge University Press |edition=reprint |year=2007 |url=https://books.google.com/books?id=Q5btdhoawN4C|isbn=978-0-521-67331-0}}</ref>

====Lower pair====
A lower pair is an ideal joint, or holonomic constraint, that maintains contact between a point, line or plane in a moving solid (three-dimensional) body to a corresponding point line or plane in the fixed solid body. There are the following cases:
* A revolute pair, or hinged joint, requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body, and a plane perpendicular to this line in the moving body maintain contact with a similar perpendicular plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom, which is pure rotation about the axis of the hinge.
* A [[prismatic joint]], or slider, requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body, and a plane parallel to this line in the moving body maintain contact with a similar parallel plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom. This degree of freedom is the distance of the slide along the line.
* A cylindrical joint requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body. It is a combination of a revolute joint and a sliding joint. This joint has two degrees of freedom. The position of the moving body is defined by both the rotation about and slide along the axis.
* A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body. This joint has three degrees of freedom.
* A planar joint requires that a plane in the moving body maintain contact with a plane in fixed body. This joint has three degrees of freedom.

====Higher pairs====
Generally speaking, a higher pair is a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. For example, the contact between a cam and its follower is a higher pair called a ''cam joint''. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.

===Kinematic chains===
[[File:Kinematics of Machinery - Figure 21.jpg|thumb|right|300px|alt=Illustration of a Four-bar linkage from Kinematics of Machinery, 1876|Illustration of a four-bar linkage from [[:s:The Kinematics of Machinery|Kinematics of Machinery, 1876]]]]
Rigid bodies ("links") connected by [[kinematic pair]]s ("joints") are known as ''[[kinematic chain]]s''. [[Mechanism (engineering)|Mechanisms]] and robots are examples of kinematic chains. The [[Degrees of freedom (mechanics)|degree of freedom]] of a kinematic chain is computed from the number of links and the number and type of joints using the [[Chebychev–Grübler–Kutzbach criterion|mobility formula]]. This formula can also be used to enumerate the [[Topology|topologies]] of kinematic chains that have a given degree of freedom, which is known as ''type synthesis'' in machine design.

====Examples====
The planar one degree-of-freedom [[Linkage (mechanical)|linkage]]s assembled from ''N'' links and ''j'' hinges or sliding joints are:
* ''N'' = 2, ''j'' = 1 : a two-bar linkage that is the lever;
* ''N'' = 4, ''j'' = 4 : the [[four-bar linkage]];
* ''N'' = 6, ''j'' = 7 : a [[six-bar linkage]]. This must have two links ("ternary links") that support three joints. There are two distinct topologies that depend on how the two ternary linkages are connected. In the [[Six-bar linkage#Watt six-bar linkage|Watt topology]], the two ternary links have a common joint; in the [[Six-bar linkage#Stephenson six-bar linkage|Stephenson topology]], the two ternary links do not have a common joint and are connected by binary links.<ref>{{cite book |last=Tsai |first=Lung-Wen |title=Mechanism design:enumeration of kinematic structures according to function |publisher=CRC Press |edition=illustrated |year=2001 |page=121 |url=https://books.google.com/books?id=X0AHKxwWTsYC&pg=PA107 |isbn=978-0-8493-0901-4}}</ref>
* ''N'' = 8, ''j'' = 10 : eight-bar linkage with 16 different topologies;
* ''N'' = 10, ''j'' = 13 : ten-bar linkage with 230 different topologies;
* ''N'' = 12, ''j'' = 16 : twelve-bar linkage with 6,856 topologies.

For larger chains and their linkage topologies, see R. P. Sunkari and [[Linda Schmidt|L. C. Schmidt]], "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm", ''Mechanism and Machine Theory'' #41, pp.&nbsp;1021–1030 (2006).

== See also ==
{{div col|colwidth=20em}}
* [[Absement]]
* [[Acceleration]]
* {{section link|Affine geometry|Kinematics}}
* [[Analytical mechanics]]
* [[Applied mechanics]]
* [[Celestial mechanics]]
* [[Centripetal force]]
* [[Classical mechanics]]
* [[Distance]]
* [[Dynamics (physics)]]
* [[Fictitious force]]
* [[Forward kinematics]]
* [[Four-bar linkage]]
* [[Inverse kinematics]]
* [[Jerk (physics)]]
* [[Kepler's laws]]
* [[Kinematic coupling]]
* [[Kinematic diagram]]
* [[Kinematic synthesis]]
* [[Kinetics (physics)]]
* [[Motion (physics)]]
* [[Orbital mechanics]]
* [[Statics]]
* [[Velocity]]
* [[Integral kinematics]]
* [[Chebychev–Grübler–Kutzbach criterion]]
{{div col end}}

== References ==
{{Reflist|30em}}
{{Reflist|group=lower-alpha}}

== Further reading ==
* {{citation |last=Koetsier |first=Teun |year=1994 |contribution=§8.3 Kinematics |pages=994–1001 |title=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences |volume=2 |editor1-last=Grattan-Guinness |editor1-first=Ivor |editor1-link=Ivor Grattan-Guinness |publisher=[[Routledge]] |isbn=0-415-09239-6}}
* {{cite book |author1-link=Francis C. Moon |last=Moon |first=Francis C. |title=The Machines of Leonardo Da Vinci and Franz Reuleaux, Kinematics of Machines from the Renaissance to the 20th Century |year=2007 |publisher=Springer |isbn=978-1-4020-5598-0}}
* [[Eduard Study]] (1913) D.H. Delphenich translator, [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/study-analytical_kinematics.pdf "Foundations and goals of analytical kinematics"].

==External links==
{{Wiktionary|kinematics}}
{{Commons category}}
*[http://www.phy.hk/wiki/englishhtm/Kinematics.htm Java applet of 1D kinematics]
*[https://web.archive.org/web/20070601020244/http://www.physclips.unsw.edu.au/ Physclips: Mechanics with animations and video clips] from the University of New South Wales.
*[http://kmoddl.library.cornell.edu/index.php Kinematic Models for Design Digital Library (KMODDL)], featuring movies and photos of hundreds of working models of mechanical systems at [[Cornell University]] and an [http://kmoddl.library.cornell.edu/e-books.php e-book library] of classic texts on mechanical design and engineering.
*[http://www.precisionballs.com/Micro_Inch_Positioning_with_Kinematic_Components.php Micro-Inch Positioning with Kinematic Components]

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[[Category:Kinematics| ]]
[[Category:Classical mechanics]]
[[Category:Mechanisms (engineering)]]