deletions | additions       diff --git a/vec_a_t_vec_a__.tex b/vec_a_t_vec_a__.tex  index e9fe7a8..f012228 100644  --- a/vec_a_t_vec_a__.tex  +++ b/vec_a_t_vec_a__.tex 
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$$(\vec a(t) + \vec a(t+\Delta t))\Delta t^2= \vec x(t-\Delta t) - \vec x(t) + \vec{x}(t+2 \Delta t) - \vec x(t+\Delta t) + O(4) $$  $$ \vec{v}(t + \Delta t) = \frac{\vec{x}(t+2\Delta t) - \vec{x}(t)}{2 \Delta t}$$  $$ \vec{v}(t) = \frac{\vec{x}(t+\Delta t) - \vec{x}(t-\Delta t)}{2 \Delta t}$$  $$(\vec $$\frac{1}{2}(\vec  a(t) + \vec a(t+\Delta t))\Delta t^2= \vec x(t-\Delta t) + t =  \vec{v}(t + \Delta t) - \vec x(t+\Delta t) \vec{v}(t)  + O(4) O(3)  $$