deletions | additions       diff --git a/vec_a_t_vec_a__.tex b/vec_a_t_vec_a__.tex  index d55b360..1e859f5 100644  --- a/vec_a_t_vec_a__.tex  +++ b/vec_a_t_vec_a__.tex 
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$$(\vec a(t) + \vec a(t+\Delta t))\Delta t^2= \vec x(t-\Delta t) - \vec x(t) + \vec{x}(t+2 \Delta t) - \vec x(t+\Delta t) $$  Taking into account that   $$ \vec{v}(t + \Delta t) = \frac{\vec{x}(t+2\Delta t) - \vec{x}(t)}{2 \Delta t} ,$$  this gets us  $$(\vec a(t) + \vec a(t+\Delta t))\Delta t^2= \vec{v}(t+\Delta t) 2 \Delta t - \vec{v}(t) 2 \Delta t $$  $$ \vec{v}(t+\Delta t) = \vec{v}(t) + \frac{1}{2}(\vec a(t) + \vec a(t+\Delta t))\Delta t $$