This is an update and expansion of the post *here*.

Here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time = *t*, location = **x**, initial location = **x**(*t*_{0}) = **x _{0}**, velocity =

**v**, initial velocity =

**v**(

*t*

_{0}) =

**v**, speed =

_{0}*v*= |

**v**|, and acceleration =

**a**.

First equation of motion

**v** = ∫ **a** d*t* = **v _{0}** +

**a**

*t*

Second equation of motion

**x** = ∫ (**v _{0}** +

**a**

*t*) d

*t*=

**x**+

_{0}**v**

_{0}*t*+ ½

**a**

*t*²

Third equation of motion

From *v*² = **v** ∙ **v** = (**v _{0}** +

**a**

*t*) ∙ (

**v**+

_{0}**a**

*t*) =

*v*

_{0}² + 2

*t*(

**a**∙

**v**) +

_{0}*a*²

*t*², and

(2**a**) ∙ (**x** ‒ **x _{0}**) = (2

**a**) ∙ (

**v**

_{0}*t*+ ½

**a**

*t*²) = 2

*t*(

**a**∙

**v**) +

_{0}*a*²

*t*² =

*v*² ‒

*v*

_{0}², it follows that

*v*² = *v*_{0}² + 2(**a** ∙ (**x** ‒ **x _{0}**)), or

*v*² − *v*_{0}² = 2**a*** ∙ ***x**, with **x _{0}** =

**0**.

Here is a derivation of the time-space equations of motion, in which retardation is constant. Let stance = *x*, time (chronation) = **t**, initial time = **t**(*x*_{0}) = **t _{0}**, lenticity =

**w**, initial lenticity =

**w**(

*x*

_{0}) =

**w**, pace

_{0}*w*= |

**w**|, and retardation =

**b**.

First equation of motion

**w** = ∫ **b** d*x* = **w _{0}** +

**b**

*x*

Second equation of motion

**t** = ∫ (**w _{0}** +

**b**

*x*) d

*x*=

**t**+

_{0}**w**

_{0}*x*+ ½

**b**

*x*²

Third equation of motion

From *w*² = **w** ∙ **w** = (**w _{0}** +

**b**

*x*) ∙ (

**w**+

_{0}**b**

*x*) =

*w*

_{0}² + 2

*x*(

**b**∙

**w**) +

_{0}*b*²

*x*², and

(2**b**) ∙ (**t** ‒ **t _{0}**) = (2

**b**) ∙ (

**w**

_{0}*x*+ ½

**b**

*x*²) = 2

*x*(

**b**∙

**w**) +

_{0}*b*²

*x*² =

*w*² ‒

*w*

_{0}², it follows that

*w*² = *w*_{0}² + 2(**b** ∙ (**t** ‒ **t _{0}**)), or

*w*² − *w*_{0}² = 2**b*** ∙ ***t**, with **t _{0}** =

**0**.

Here is a derivation of the space-time *weighted* equations of motion with weight *m* (mass), in which acceleration is constant. Let time = *t*, weighted location = **ξ** = *m***x**, initial weighted location = **ξ**(*t*_{0}) = *m***x _{0}**, momentum =

**p**=

*m*

**v**, initial momentum =

**p**(

*t*

_{0}) =

**p**=

_{0}*m*

**v**, scalar momentum

_{0}*p*= |

**p**|, and weighted acceleration (force) =

**F**=

*m*

**a**.

First weighted equation of motion

**p** = ∫ **F** d*t* = **p _{0}** +

**F**

*t*

Second weighted equation of motion

**ξ** = ∫ (**p _{0}** +

**F**

*t*) d

*t*=

**ξ**+

_{0}**p**

_{0}*t*+ ½

**F**

*t*²

Third weighted equation of motion

*mv*² − *mv*_{0}² = 2m**a*** ∙ ***x** = 2**F*** ∙ ***x**

½*mv*² − ½*mv*_{0}² = m**a*** ∙ ***x** = **F*** ∙ ***x** = ΔKE = W,

where KE is the *kinetic energy* and W is the *work*.

Here is a derivation of the time-space *weighted* equations of motion with weight *n* (vass), in which retardation is constant. Let stance = *x*, weighted time (chronation) = **τ** = *n***t**, initial weighted chronation = **τ**(*x*_{0}) = *n***t _{0}**, fulmentum =

**q**=

*n*

**w**, initial fulmentum =

**q**(

*x*

_{0}) =

**q**=

_{0}*n*

**w**, scalar fulmentum

_{0}*q*= |

**q**|, and weighted retardation (release) =

**R**=

*n*

**b**.

First weighted equation of motion

**q** = ∫ **R** d*x* = **q _{0}** +

**R**

*x*

Second weighted equation of motion

**τ** = ∫ (**q _{0}** +

**R**

*x*) d

*x*=

**τ**+

_{0}**q**

_{0}*x*+ ½

**R**

*x*²

Third weighted equation of motion

*nw*² − n*w*_{0}² = 2n**b*** ∙ ***t** = 2**R*** ∙ ***t**

½*nw*² − ½*nw*_{0}² = n**b*** ∙ ***t** = **R*** ∙ ***t** = ΔKL = Z,

where KL is the *kinetic lethargy* and Z is the *repose* (inverse of work).