First, here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time (distimement) = *t*, position = **r**, initial position = **r**(*t*_{0}) = **r _{0}**, velocity =

**v**, initial velocity =

**v**(

*t*

_{0}) =

**v**,

_{0}*v*= |

**v**| = speed, and acceleration =

**a**.

First equation of motion

**v** = ∫ **a** d*t* = **v _{0}** +

**a**

*t*

Second equation of motion

**r** = ∫ (**v _{0}** +

**a**

*t*) d

*t*=

**r**+

_{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**) ∙ (**r** ‒ **r _{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** ∙ (**r** ‒ **r _{0}**)).

Next, here is a derivation of the time-space equations of motion, in which modulation is constant. Let position (displacement) = *r*, time = **t**, initial time = **t**(*r*_{0}) = **t _{0}**, legerity =

**u**, initial legerity =

**u**(

*r*

_{0}) =

**u**,

_{0}*u*= |

**u**| = pace, and modulation =

**b**.

First equation of motion

**u** = ∫ **b** d*r* = **u _{0}** +

**b**

*t*

Second equation of motion

**t** = ∫ (**u _{0}** +

**b**

*r*) d

*r*=

**t**+

_{0}**u**

_{0}*r*+ ½

**b**

*r*²

Third equation of motion

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

**b**

*r*) ∙ (

**u**+

_{0}**b**

*r*) =

*u*

_{0}² + 2

*r*(

**b**∙

**u**) +

_{0}*b*²

*r*², and

(2**b**) ∙ (**t** ‒ **t _{0}**) = (2

**b**) ∙ (

**u**

_{0}*r*+ ½

**b**

*r*²) = 2

*r*(

**b**∙

**u**) +

_{0}*b*²

*r*² =

*u*² ‒

*u*

_{0}², it follows that

*u*² = *u*_{0}² + 2(**b** ∙ (**t** ‒ **t _{0}**)).