[Contexto]
( 1 ) v = d r d t . {\displaystyle (1)\quad v={\frac {\mathrm {d} r}{\mathrm {d} t}}\quad .} | Definición de Velocidad
( 2 a ) a = d v d t = c t e . ( 2 b ) a = d 2 r d t 2 = c t e . . {\displaystyle {\begin{matrix}(2a)&a&=&{\frac {\mathrm {d} v}{\mathrm {d} t}}=\mathrm {cte.} \\(2b)&a&=&{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}=\mathrm {cte.} \end{matrix}}\quad .} | Definición de Aceleración
( I ) v = ∫ a d t = a t + v 0 . {\displaystyle (\mathrm {I} )\quad v=\!\int \!{a\,\mathrm {d} t}=a\,t+v_{0}\quad .} | Integrando (2a)
( I I ) r = ∫ ∫ a d t 2 = 1 2 a t 2 + v 0 t + r 0 . {\displaystyle (\mathrm {II} )\quad r=\!\int \!\!\!\int \!{a\,\mathrm {d} t^{2}}={\frac {1}{2}}a\,t^{2}+v_{0}\,t+r_{0}\quad .} | Integrando (2b)
( I I I ) a = v − v 0 t d e ( I ) {\displaystyle (\mathrm {III} )\quad a={\frac {v-v_{0}}{t}}\quad \mathrm {de\ (I)} }
( I V ) r = 1 2 v − v 0 t t 2 + v 0 t + r 0 d e ( I I ) y ( I I I ) = 1 2 ( v − v 0 ) t + v 0 t + r 0 = 1 2 ( v + v 0 ) t + r 0 {\displaystyle {\begin{matrix}(\mathrm {IV} )\quad r&=&{\frac {1}{2}}{\frac {v-v_{0}}{t}}t^{2}+v_{0}\,t+r_{0}&\quad \mathrm {de\ (II)\ y\ (III)} \\&=&{\frac {1}{2}}(v-v_{0})t+v_{0}\,t+r_{0}\\&=&{\frac {1}{2}}(v+v_{0})t+r_{0}\end{matrix}}}
( V ) r − r 0 = 1 2 ( v + v 0 ) t d e ( I V ) {\displaystyle (\mathrm {V} )\quad r-r_{0}={\frac {1}{2}}(v+v_{0})t\quad \mathrm {de\ (IV)} }
( V I ) 2 ( r − r 0 ) = ( v + v 0 ) t d e ( V ) . {\displaystyle (\mathrm {VI} )\quad 2(r-r_{0})=(v+v_{0})t\quad \mathrm {de\ (V)} \quad .} | C.q.d.
( V I I ) t = v − v 0 a d e ( I ) {\displaystyle (\mathrm {VII} )\quad t={\frac {v-v_{0}}{a}}\quad \mathrm {de\ (I)} }
( V I I I ) r = 1 2 a ( v − v 0 a ) 2 + v 0 v − v 0 a + r 0 d e ( I I ) y ( V I I ) {\displaystyle (\mathrm {VIII} )\quad r={\frac {1}{2}}a\left({\frac {v-v_{0}}{a}}\right)^{2}+v_{0}{\frac {v-v_{0}}{a}}+r_{0}\quad \mathrm {de\ (II)\ y\ (VII)} }
( I X ) r − r 0 = 1 2 a ( v − v 0 a ) 2 + v 0 v − v 0 a d e ( V I I I ) {\displaystyle (\mathrm {IX} )\quad r-r_{0}={\frac {1}{2}}a\left({\frac {v-v_{0}}{a}}\right)^{2}+v_{0}{\frac {v-v_{0}}{a}}\quad \mathrm {de\ (VIII)} }
( X ) 2 a ( r − r 0 ) = ( v − v 0 ) 2 + 2 v 0 ( v − v 0 ) d e ( I X ) = ( v − v 0 ) [ ( v − v 0 ) + 2 v 0 ] = ( v − v 0 ) ( v + v 0 ) {\displaystyle {\begin{matrix}(\mathrm {X} )\quad 2\,a(r-r_{0})&=&(v-v_{0})^{2}+2\,v_{0}(v-v_{0})&\quad \mathrm {de\ (IX)} \\&=&(v-v_{0})\left[(v-v_{0})+2\,v_{0}\right]\\&=&(v-v_{0})(v+v_{0})\\\end{matrix}}}
( X I ) 2 a ( r − r 0 ) = v 2 − v 0 2 d e ( X ) . {\displaystyle (\mathrm {XI} )\quad 2\,a(r-r_{0})=v^{2}-{v_{0}}^{2}\quad \mathrm {de\ (X)} \quad .} | C.q.d.
| Definición de Velocidad
| Definición de Aceleración
| Integrando (2a)
| Integrando (2b)
| C.q.d.
![{\displaystyle {\begin{matrix}(\mathrm {X} )\quad 2\,a(r-r_{0})&=&(v-v_{0})^{2}+2\,v_{0}(v-v_{0})&\quad \mathrm {de\ (IX)} \\&=&(v-v_{0})\left[(v-v_{0})+2\,v_{0}\right]\\&=&(v-v_{0})(v+v_{0})\\\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be72407f88877201080745ea64c408fda90f2645)
| C.q.d.
