deletions | additions       diff --git a/If_we_assume_2_vec__.tex b/If_we_assume_2_vec__.tex  index ba65cfe..0effcf0 100644  --- a/If_we_assume_2_vec__.tex  +++ b/If_we_assume_2_vec__.tex 
...
If we assume  $$2  \vec{v}(t)\Delta t  = \vec{x}(t+\Delta \frac{\vec{x}(t+\Delta  t) - \vec{x}(t-\Delta t) t)}{2 \Delta t}  $$ then  $$\vec x(t+\Delta t)=2\vec x(t)-\vec x(t-\Delta t)+\vec a(t)\,\Delta t^2$$ $$ \vec{v}(t) = \frac{\vec{x}(t+\Delta t) - \vec{x}(t) + \vec{x}(t) - \vec{x}(t-\Delta t)}{2 \Delta t} $$  $$\vec x(t+\Delta t)=\vec x(t) $$ \vec{v}(t) = \frac{1}{2} (\frac{\vec{x}(t+\Delta t) - \vec{x}(t)}{\Delta t}  + (\vec x(t)-\vec x(t-\Delta t)+\frac{1}{2} \vec a(t) \frac{\vec{x}(t) - \vec{x}(t-\Delta t)}{\Delta t} ) $$  $$ \vec{v}(t+\Delta t) = \frac{\vec{x}(t+2  \Delta t^2) + \frac{1}{2} \vec a(t) t) - \vec{x}(t)}{2  \Delta t^2$$ t} $$