# Equations of motion

The first two equations of motion are

$v→av=Δd→Δt(𝟏)$ and $a→av=Δv→Δt(𝟐).$

If we substitute $\mathrm{\Delta }x={x}_{2}-{x}_{1}$ for $Δd→$ and $Δv→,$ we get

$d→2=d→1+v→avΔt(𝟑)$ and $v→2=v→1+a→avΔt(𝟒).$

If we take $Δd→=v→avΔt$ and substitute the average of two velocities (initial and final) for $v→av,$ we get

$Δd→=v→1+v→22Δt(𝟓).$

We can substitute equation (4) into this, yielding

$Δd→=v→1+(v→1+a→avΔt)2Δt=2v→1+a→avΔt2Δt,$

which simplifies to

$Δd→=v→1Δt+12a→av(Δt)2(𝟔).$

If we rearrange equation (4) to isolate $v→1$ and then substitute that into equation (5), we get

$Δd→=v→2Δt−12a→av(Δt)2(𝟕).$

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 $\mathrm{\Delta }t\text{,}$ and then we substitute that into equation (5), giving us

$Δd→=(v→1+v→22)(v→2−v→1a→av).$

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

$2a→avΔd→=v→22−v→12,$

which we can rearrange to get our final equation,

$v→22=v→12+2a→avΔd→(𝟖).$

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.