this is for holding javascript data
Ghlen Livid added Now_let_s_derive_x__.tex
almost 8 years ago
Commit id: e5d53d06a018e827d84993dda95655592736edac
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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 can see that
$$\vec{v}(t- \Delta t) \Delta t + \frac{1}{2} \vec{a}(t- \Delta t) \Delta t^2 = \vec{x}(t) - \vec{x}(t - \Delta t)$$
Then (6) turns into
$$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{x}(t) - \vec{x}(t - \Delta t) + \vec{a}(t) \Delta t^2$$
$$\vec{x}(t + \Delta t) = 2 \vec{x}(t) - \vec{x}(t - \Delta t) + \vec{a}(t) \Delta t^2$$
which is \textit{exactly} (3). $\not{\Delta}$
