I know the equation for displacement as a function of time is $$\vec{s} = \vec{v_i} \Delta{t}+\frac{1}{2}\vec{a}\,(\Delta t)^2$$
I need to solve for $\Delta t$ I'm having problems rearranging to do this. As of now, I created a Python program to run through the displacement formula, adding a set amount of time in each iteration, until the displacement is equal to the displacement I am looking for. I feel that this is woefully inefficient, inaccurate, and horribly CPU intensive. I am confident that this is possible to rearrange, and that I have missed something.
If someone already know what this formula rearranged to solve for $\Delta t$ is, I'd much appreciate it if you would share it with me. In the meantime, I shall continue to look around for the rearranged formula and keep trying to solve for $\Delta t$.
Hopefully I have not missed something painfully obvious, and that I am not wasting anyone's time.
Best Answer
If you truly have a vector equation, then you really have three quadratic equations - one each for the X, Y and Z component.
Let's write them:
$$s_x = v_x \Delta t + \frac12 a_x (\Delta t)^2\\ s_y = v_y \Delta t + \frac12 a_y (\Delta t)^2\\ s_z = v_z \Delta t + \frac12 a_z (\Delta t)^2$$
If there is only one value of $\Delta t$, then this is an overdetermined set of equations: three equations, one unknown. This means that the values of acceleration and velocity have to be exactly right if you want to be able to solve them.
But assuming that this is the case, then the solution (after simply rearranging) is the solution to the quadratic equation:
$$\frac12 a t^2 + v t - s = 0$$ (I dropped some suffixes to make it easier to write). We solve for $t$ with the usual:
$$t = \frac{-v \pm \sqrt{v^2 + 2as}}{a}$$