Editing Equations of motion (section)

===Uniform acceleration===

The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below.

====Constant translational acceleration in a straight line====<!--'SUVAT equations' and 'Suvat equations' redirect here-->
<!---THERE ARE NUMEROUS NAMES - PLEASE LEAVE THESE EQUATIONS AS "SUVAT" --->
These equations apply to a particle moving linearly, in three dimensions in a straight line with constant [[acceleration]].<ref name="Physics P.M">{{cite book | last1 = Whelan | first1 = P. M. | last2 = Hodgson | first2 = M. J. | url = https://www.worldcat.org/oclc/7102249 | title = Essential Principles of Physics | date = 1978 | publisher = John Murray | edition = second | isbn = 0-7195-3382-1 | location = London | oclc = 7102249}}</ref> Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one.

<math display="block">\begin{align}
v & = v_0 + a t & [1]\\
r & = r_0 + v_0 t + \tfrac12 {a}t^2 & [2]\\
r & = r_0 + \tfrac12 \left( v+v_0 \right )t & [3]\\
v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4]\\
r & = r_0 + vt - \tfrac12 {a}t^2 & [5]\\
\end{align}</math>

where:
*{{math|''r''<sub>0</sub>}} is the particle's initial [[Position (vector)|position]]
*{{math|''r''}} is the particle's final position
*{{math|''v''<sub>0</sub>}} is the particle's initial [[velocity]]
*{{math|''v''}} is the particle's final velocity
*{{math|''a''}} is the particle's [[acceleration]]
*{{math|''t''}} is the [[Time in physics|time interval]]

{{hidden begin
|toggle     = left
|title      = Derivation
|titlestyle = text-align:right; font-size:100%; padding-right:2.5em;
|contentstyle = text-align:left; font-size:100%; padding-left:1.5em;
}}

Equations [1] and [2] are from integrating the definitions of velocity and acceleration,<ref name="Physics P.M"/> subject to the initial conditions {{math|'''r'''(''t''<sub>0</sub>) {{=}} '''r'''<sub>0</sub>}} and {{math|'''v'''(''t''<sub>0</sub>) {{=}} '''v'''<sub>0</sub>}};

<math display="block">\begin{align}
\mathbf{v} & = \int \mathbf{a} dt = \mathbf{a}t+\mathbf{v}_0 \,, & [1] \\ 
\mathbf{r} & = \int (\mathbf{a}t+\mathbf{v}_0) dt = \frac{\mathbf{a}t^2}{2}+\mathbf{v}_0t +\mathbf{r}_0 \,, & [2] \\
\end{align}</math>

in magnitudes,

<math display="block">\begin{align}
v & = at+v_0 \,, & [1] \\ 
r & = \frac{{a}t^2}{2}+v_0t +r_0 \,. & [2] \\
\end{align}</math>

Equation [3] involves the average velocity {{math|{{sfrac|'''v''' + '''v'''<sub>0</sub>|2}}}}. Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from {{math|'''v'''<sub>0</sub>}} to {{math|'''v'''}}, as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows from solving [1] for

<math display="block" qid=Q11376> \mathbf{a} = \frac{(\mathbf{v} - \mathbf{v}_0)}{t} </math>

and substituting into [2]

<math display="block"> \mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{t}{2}(\mathbf{v} - \mathbf{v}_0) \,, </math>

then simplifying to get

<math display="block"> \mathbf{r} = \mathbf{r}_0 + \frac{t}{2}(\mathbf{v} + \mathbf{v}_0) </math>

or in magnitudes

<math display="block"> r = r_0 + \left( \frac{v+v_0}{2} \right )t \quad [3] </math>

From [3],

<math display="block">t = \left( r - r_0 \right)\left( \frac{2}{v+v_0} \right )</math>

substituting for {{mvar|t}} in [1]:

<math display="block">\begin{align}
v & = a\left( r - r_0 \right)\left( \frac{2}{v+v_0} \right )+v_0 \\
v\left( v+v_0 \right ) & = 2a\left( r - r_0 \right)+v_0\left( v+v_0 \right ) \\
v^2+vv_0 & = 2a\left( r - r_0 \right)+v_0v+v_0^2 \\
v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4] \\
\end{align}</math>

From [3],

<math display="block"> 2\left(r - r_0\right) - vt = v_0 t </math>

substituting into [2]:

<math display="block"> \begin{align}
r & = \frac{{a}t^2}{2} + 2r - 2r_0 - vt + r_0 \\
0 & = \frac{{a}t^2}{2}+r - r_0 - vt \\
r & = r_0 + vt - \frac{{a}t^2}{2} & [5] 
\end{align}</math>

Usually only the first 4 are needed, the fifth is optional.

{{hidden end}}

Here {{math|''a''}} is ''constant'' acceleration, or in the case of bodies moving under the influence of [[gravity]], the [[standard gravity]] {{math|''g''}} is used. Note that each of the equations contains four of the five variables, so in this situation it is sufficient to know three out of the five variables to calculate the remaining two.

In some programs, such as the [[International General Certificate of Secondary Education|IGCSE]] Physics and [[IB Diploma Programme|IB DP]] [[IB Group 4 subjects|Physics]] programs (international programs but especially popular in the UK and Europe), the same formulae would be written with a different set of preferred variables. There {{math|''u''}} replaces {{math|''v''<sub>0</sub>}} and {{math|''s''}} replaces {{math|''r'' - ''r''<sub>0</sub>}}. They are often referred to as the '''SUVAT equations'''<!--boldface per WP:R#PLA-->, where "SUVAT" is an [[acronym]] from the variables: {{math|''s''}} = displacement, {{math|''u''}} = initial velocity, {{math|''v''}} = final velocity, {{math|''a''}} = acceleration, {{math|''t''}} = time.<ref name=Hanrahan2003>{{cite book |last1=Hanrahan |first1=Val |last2=Porkess |first2=R |year=2003 |title=Additional Mathematics for OCR |publisher=Hodder & Stoughton |location=London |isbn=0-340-86960-7 |page=219}}</ref><ref>{{cite book |last=Johnson |first=Keith |year=2001 |title=Physics for you: revised national curriculum edition for GCSE |edition=4th |publisher=Nelson Thornes |isbn=978-0-7487-6236-1 |page=135 |url=https://books.google.com/books?id=D4nrQDzq1jkC&q=suvat&pg=PA135 |quote=The 5 symbols are remembered by "suvat". Given any three, the other two can be found.}}</ref> In these variables, the equations of motion would be written

<math display="block">\begin{align}
v & = u + at & [1] \\
s & = ut + \tfrac12 at^2 & [2] \\
s & = \tfrac{1}{2}(u + v)t & [3] \\
v^2 & = u^2 + 2as & [4] \\
s & = vt - \tfrac12 at^2 & [5] \\
\end{align}</math>

====Constant linear acceleration in any direction====

[[File:Suvat eom any direction constant acceleration.svg|400px|thumb|Trajectory of a particle with initial position vector {{math|'''r'''<sub>0</sub>}} and velocity {{math|'''v'''<sub>0</sub>}}, subject to constant acceleration {{math|'''a'''}}, all three quantities in any direction, and the position {{math|'''r'''(''t'')}} and velocity {{math|'''v'''(''t'')}} after time {{math|''t''}}.]]

<!---THERE ARE NUMEROUS NAMES - PLEASE LEAVE THESE EQUATIONS AS "SUVAT" --->
The initial position, initial velocity, and acceleration vectors need not be collinear, and the equations of motion take an almost identical form. The only difference is that the square magnitudes of the velocities require the [[dot product]]. The derivations are essentially the same as in the collinear case,

<math display="block">\begin{align}
\mathbf{v} & = \mathbf{a}t+\mathbf{v}_0 & [1]\\
\mathbf{r} & = \mathbf{r}_0 + \mathbf{v}_0 t + \tfrac12\mathbf{a}t^2 & [2]\\
\mathbf{r} & = \mathbf{r}_0 + \tfrac12 \left(\mathbf{v}+\mathbf{v}_0\right) t & [3]\\
\mathbf{v}^2 & = \mathbf{v}_0^2 + 2\mathbf{a}\cdot\left( \mathbf{r} - \mathbf{r}_0 \right) & [4]\\
\mathbf{r} & = \mathbf{r}_0 + \mathbf{v}t - \tfrac12\mathbf{a}t^2 & [5]\\
\end{align}</math>
although the [[Torricelli equation]] [4] can be derived using the [[distributive property]] of the dot product as follows:
<math display="block">v^{2} = \mathbf{v}\cdot\mathbf{v} = (\mathbf{v}_0+\mathbf{a}t)\cdot(\mathbf{v}_0+\mathbf{a}t) = v_0^{2}+2t(\mathbf{a}\cdot\mathbf{v}_0)+a^{2}t^{2}</math>
<math display="block">(2\mathbf{a})\cdot(\mathbf{r}-\mathbf{r}_0) = (2\mathbf{a})\cdot\left(\mathbf{v}_0t+\tfrac{1}{2}\mathbf{a}t^{2}\right)=2t(\mathbf{a}\cdot\mathbf{v}_0)+a^{2}t^{2} = v^{2} - v_0^{2}</math>
<math display="block">\therefore v^{2} = v_0^{2} + 2(\mathbf{a}\cdot(\mathbf{r}-\mathbf{r}_0))</math>

====Applications====

Elementary and frequent examples in kinematics involve [[projectile]]s, for example a ball thrown upwards into the air. Given initial velocity {{math|''u''}}, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity {{math|''g''}}. While these quantities appear to be [[scalar (physics)|scalars]], the direction of displacement, speed and acceleration is important. They could in fact be considered as unidirectional vectors. Choosing {{math|''s''}} to measure up from the ground, the acceleration {{math|''a''}} must be in fact {{math|''−g''}}, since the force of [[gravity]] acts downwards and therefore also the acceleration on the ball due to it.

At the highest point, the ball will be at rest: therefore {{math|''v'' {{=}} 0}}. Using equation [4] in the set above, we have:

<math display="block">s= \frac{v^2 - u^2}{-2g}.</math>

Substituting and cancelling minus signs gives:

<math display="block">s = \frac{u^2}{2g}.</math>

====Constant circular acceleration====

The analogues of the above equations can be written for [[rotation]]. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary,

<math display="block">\begin{align}
\omega & = \omega_0 + \alpha t \\
\theta &= \theta_0 + \omega_0t + \tfrac12\alpha t^2 \\
\theta & = \theta_0 + \tfrac12(\omega_0 + \omega)t \\
\omega^2 & = \omega_0^2 + 2\alpha(\theta - \theta_0) \\
\theta & = \theta_0 + \omega t - \tfrac12\alpha t^2 \\
\end{align}</math>

where {{math|''α''}} is the constant [[angular acceleration]], {{math|''ω''}} is the [[angular velocity]], {{math|''ω''<sub>0</sub>}} is the initial angular velocity, {{math|''θ''}} is the angle turned through ([[angular displacement]]), {{math|''θ''<sub>0</sub>}} is the initial angle, and {{math|''t''}} is the time taken to rotate from the initial state to the final state.

===General planar motion===
{{hatnote|Main article: [[Centripetal force#General planar motion|General planar motion]]}}
{{multiple image
|align = vertical
|width1 = 100
|image1 = Position vector plane polar coords.svg
|caption1 = Position vector {{math|'''r'''}}, always points radially from the origin.
|width2 = 150
|image2 = Velocity vector plane polar coords.svg
|caption2 = Velocity vector {{math|'''v'''}}, always tangent to the path of motion.
|width3 = 200
|image3 = Acceleration vector plane polar coords.svg
|caption3 = Acceleration vector {{math|'''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 2D space, but a plane in any higher dimension.}}

These are the kinematic equations for a particle traversing a path in a plane, described by position {{math|'''r''' {{=}} '''r'''(''t'')}}.<ref>{{cite book | last = Halpern | first = Alvin M. | url = https://www.worldcat.org/oclc/27398318 | title = 3000 Solved Problems in Physics | date = 1988 | isbn = 978-0-07-025734-4 | location = New York | oclc = 27398318 | series = Schaum Series | publisher = McGraw Hill}}</ref> They are simply the time derivatives of the position vector in plane [[polar coordinates]] using the definitions of physical quantities above for angular velocity {{math|''ω''}} and angular acceleration {{math|''α''}}. These are instantaneous quantities which change with time.

The position of the particle is

<math display="block"> \mathbf{r} =\mathbf{r}\left ( r(t),\theta(t) \right ) = r \mathbf{\hat{e}}_r </math>

where {{math|'''ê'''{{sub|''r''}}}} and {{math|'''ê'''{{sub|''θ''}}}} are the [[Polar coordinate system#Vector calculus|polar]] unit vectors. Differentiating with respect to time gives the velocity

<math display="block">\mathbf{v} = \mathbf{\hat{e}}_r \frac{d r}{dt} + r \omega \mathbf{\hat{e}}_\theta </math>

with radial component {{math|{{sfrac|''dr''|''dt''}}}} and an additional component {{math|''rω''}} due to the rotation. Differentiating with respect to time again obtains the acceleration 

<math display="block">\mathbf{a} =\left ( \frac{d^2 r}{dt^2} - r\omega^2\right )\mathbf{\hat{e}}_r + \left ( r \alpha + 2 \omega \frac{dr}{dt} \right )\mathbf{\hat{e}}_\theta  </math>

which breaks into the radial acceleration {{math|{{sfrac|''d''<sup>2</sup>''r''|''dt''<sup>2</sup>}}}}, [[centripetal acceleration]] {{math|–''rω''<sup>2</sup>}}, [[Coriolis acceleration]] {{math|2''ω''{{sfrac|''dr''|''dt''}}}}, and angular acceleration {{math|''rα''}}.

Special cases of motion described by these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.

{| class="wikitable"
|-
!scope="col" width="20%"| '''State of motion'''
!scope="col" width="20%"| Constant {{math|''r''}}
!scope="col" width="20%"| {{math|''r''}} linear in {{math|''t''}}
!scope="col" width="20%"| {{math|''r''}} quadratic in {{math|''t''}}
!scope="col" width="20%"| ''r'' non-linear in ''t''
|-
! Constant {{math|''θ''}}
|| Stationary
|| Uniform translation (constant translational velocity)
|| Uniform translational acceleration
|| Non-uniform translation
|-
! {{math|''θ''}} linear in {{math|''t''}}
|| Uniform angular motion in a circle (constant angular velocity)
|| Uniform angular motion in a spiral, constant radial velocity
|| Angular motion in a spiral, constant radial acceleration
|| Angular motion in a spiral, varying radial acceleration
|-
!{{math|''θ''}} quadratic in {{math|''t''}}
|| Uniform angular acceleration in a circle
|| Uniform angular acceleration in a spiral, constant radial velocity
|| Uniform angular acceleration in a spiral, constant radial acceleration
|| Uniform angular acceleration in a spiral, varying radial acceleration
|-
! {{math|''θ''}} non-linear in {{math|''t''}}
|| Non-uniform angular acceleration in a circle
|| Non-uniform angular acceleration in a spiral, constant radial velocity
|| Non-uniform angular acceleration in a spiral, constant radial acceleration
|| Non-uniform angular acceleration in a spiral, varying radial acceleration
|-
|}