deletions | additions       diff --git a/Now_let_s_derive_x__.tex b/Now_let_s_derive_x__.tex  index 4651e82..9fb9f61 100644  --- a/Now_let_s_derive_x__.tex  +++ b/Now_let_s_derive_x__.tex 
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Now, let's derive $x(t)$ from (1) by time-shifting:  $$\vec{x}(t) = \vec{x}(t - \Delta t) + \vec{v}(t- \Delta t) \Delta t + \frac{1}{2} \vec{a}(t- \Delta t) \Delta t^2$$  We From (5), we  can see that $$\vec{v}(t- \Delta t) \Delta t + \frac{1}{2} \vec{a}(t- \Delta t) \Delta t^2 = \vec{x}(t) - \vec{x}(t - \Delta t)$$