deletions | additions       diff --git a/vec_v_t_vec_v__.tex b/vec_v_t_vec_v__.tex  index 404b9e8..7a24028 100644  --- a/vec_v_t_vec_v__.tex  +++ b/vec_v_t_vec_v__.tex 
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$$\vec{x}(t + \Delta t) = \vec{x}(t) + (\vec{v}(t - \Delta t) + \frac{\vec{a}(t - \Delta t) + \vec{a}(t)}{2} \Delta t) \Delta t + \frac{1}{2} \vec{a}(t) \Delta t^2$$  $$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t - \Delta t) \Delta t + \frac{\vec{a}(t - \Delta t)}{2}+ \frac{\vec{a}(t)}{2}  \Delta t^2 +\frac{1}{2}  \vec{a}(t) \Delta t^2$$