Let’s reformulate (1) and (2) for \(t = t - \Delta t\)

\[\vec{v}(t) = \vec{v}(t - \Delta t) + \frac{\vec{a}(t - \Delta t) + \vec{a}(t)}{2} \Delta t\] \[\vec{x}(t) = \vec{x}(t - \Delta t) + \vec{v}(t - \Delta t) \Delta t + \frac{1}{2} \vec{a}(t - \Delta t) \Delta t^2\]