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**_{0}, speed = *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**_{0}, pace *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**.

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