Equations of motion

The first two equations of motion are

vav=ΔdΔt(𝟏) and aav=ΔvΔt(𝟐).

If we substitute Δx=x2x1 for Δd and Δv, we get

d2=d1+vavΔt(𝟑) and v2=v1+aavΔt(𝟒).

If we take Δd=vavΔt and substitute the average of two velocities (initial and final) for vav, we get


We can substitute equation (4) into this, yielding


which simplifies to


If we rearrange equation (4) to isolate v1 and then substitute that into equation (5), we get


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 Δt, and then we substitute that into equation (5), giving us


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


which we can rearrange to get our final equation,


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.