Equations of motion

The first two equations of motion are

`vec v_"av" = (Delta vec d)/(Delta t)quad`(1)`qquad and qquad vec a_"av" = (Delta vec v)/(Delta t)quad`(2).

If we substitute `Delta x = x_2 - x_1` for `Delta vec d` and `Delta vec v`, we get

`vec d_2 = vec d_1 + vec v_"av"Delta tquad`(3)`qquad and qquad vec v_2 = vec v_1 + vec a_"av"Delta tquad`(4).

If we take `Delta vec d = vec v_"av"Delta t` and substitute the average of two velocities (initial and final) for `vec v_"av"`, we get

`Delta vec d = (vec v_1 + vec v_2)/2Delta tquad`(5).

We can substitute equation (4) into this, yielding

`Delta vec d = (vec v_1 + (vec v_1 + vec a_"av"Delta t))/2Delta t = (2vec v_1 + vec a_"av"Delta t)/2Delta t`,

which simplifies to

`Delta vec d = vec v_1Delta t + 1/2vec a_"av"(Delta t)^2quad`(6).

If we rearrange equation (4) to isolate `vec v_1` and then substitute that into equation (5), we get

`Delta vec d = vec v_2Delta t - 1/2vec a_"av"(Delta t)^2quad`(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 `Delta t`, and then we substitute that into equation (5), giving us

`Delta vec d = ((vec v_1 + vec v_2)/2)((vec v_2 - vec v_1)/vec a_"av")`.

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

`2vec a_"av"Delta vec d = vec v_2^2 - vec v_1^2`,

which we can rearrange to get our final equation,

`vec v_2^2 = vec v_1^2 + 2vec a_"av"Delta vec dquad`(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.