deletions | additions       diff --git a/Now_vec_v_t_Delta__.tex b/Now_vec_v_t_Delta__.tex  index 4d5f867..fb7f472 100644  --- a/Now_vec_v_t_Delta__.tex  +++ b/Now_vec_v_t_Delta__.tex 
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Now $$ \vec{v}(t) \Delta t = \frac{1}{2} (\vec{x}(t+\Delta t) - \vec x(t-\Delta t)) $$  equivalently, (11) can be equivalently written as,  $$ \vec{v}(t) \Delta t = \vec{x}(t) - \vec{x}(t-\Delta t) + \frac{1}{2} (\vec{x}(t+\Delta t) - 2\vec x(t) + \vec x(t-\Delta t)) $$  But  $$\vec a(t)\Delta t^2=\vec{x}(t+\Delta t) - 2\vec x(t) + \vec x(t-\Delta t)$$ Now we can substitute (12)  so that  $$ \vec{v}(t) \Delta t = \vec{x}(t) - \vec{x}(t-\Delta t) + \frac{1}{2} \vec{a}(t) \Delta t^2 $$