# Equations of motion

The first two equations of motion are

$\displaystyle \vec{v}_{\text{av}} = \frac{\Delta{} \vec{d}}{\Delta{} t} \quad$(1)$\displaystyle \qquad \allowbreak\quad\allowbreak\text{and}\allowbreak\quad\allowbreak \qquad \vec{a}_{\text{av}} = \frac{\Delta{} \vec{v}}{\Delta{} t} \quad$(2).

If we substitute $\displaystyle \Delta{} x = x_{2} - x_{1}$ for $\displaystyle \Delta{} \vec{d}$ and $\displaystyle \Delta{} \vec{v}$, we get

$\displaystyle \vec{d}_{2} = \vec{d}_{1} + \vec{v}_{\text{av}} \Delta{} t \quad$(3)$\displaystyle \qquad \allowbreak\quad\allowbreak\text{and}\allowbreak\quad\allowbreak \qquad \vec{v}_{2} = \vec{v}_{1} + \vec{a}_{\text{av}} \Delta{} t \quad$(4).

If we take $\displaystyle \Delta{} \vec{d} = \vec{v}_{\text{av}} \Delta{} t$ and substitute the average of two velocities (initial and final) for $\displaystyle \vec{v}_{\text{av}}$, we get

$\displaystyle \Delta{} \vec{d} = \frac{\vec{v}_{1} + \vec{v}_{2}}{2} \Delta{} t \quad$(5).

We can substitute equation (4) into this, yielding

$\displaystyle \Delta{} \vec{d} = \frac{\vec{v}_{1} + \left ( \vec{v}_{1} + \vec{a}_{\text{av}} \Delta{} t \right )}{2} \Delta{} t = \frac{2 \vec{v}_{1} + \vec{a}_{\text{av}} \Delta{} t}{2} \Delta{} t$,

which simplifies to

$\displaystyle \Delta{} \vec{d} = \vec{v}_{1} \Delta{} t + \frac{1}{2} \vec{a}_{\text{av}} \left ( \Delta{} t \right )^{2} \quad$(6).

If we rearrange equation (4) to isolate $\displaystyle \vec{v}_{1}$ and then substitute that into equation (5), we get

$\displaystyle \Delta{} \vec{d} = \vec{v}_{2} \Delta{} t - \frac{1}{2} \vec{a}_{\text{av}} \left ( \Delta{} t \right )^{2} \quad$(7).

We can derive one final equation, this time eliminating the one variable that has been present in all the others: time. We begin by rearranging equation (4) to isolate $\displaystyle \Delta{} t$, and then we substitute that into equation (5), giving us

$\displaystyle \Delta{} \vec{d} = \left ( \frac{\vec{v}_{1} + \vec{v}_{2}}{2} \right ) \left ( \frac{\vec{v}_{2} - \vec{v}_{1}}{\vec{a}_{\text{av}}} \right )$.

By multiplying the denominators to the other side and recognizing the difference of squares, we get

$\displaystyle 2 \vec{a}_{\text{av}} \Delta{} \vec{d} = \vec{v}_{2}^{2} - \vec{v}_{1}^{2}$,

which we can rearrange to get our final equation,

$\displaystyle \vec{v}_{2}^{2} = \vec{v}_{1}^{2} + 2 \vec{a}_{\text{av}} \Delta{} \vec{d} \quad$(8).

There might be a few more equations that we could have derived, but these eight (and rearranged versions of them) should take you a long way.