Special Trajectory Equation - Solving for T

[Zengi]

Hello everyone,
I am making a physics model for a game. This model works by extrapolating points in time through trajectories, pretty basic stuff. The problem is I need a trajectory for something on a 2d plane, with constant acceleration, and constant angular velocity. I've done almost all the leg work but I got stuck at the end when I need to solve it for t. I need to solve for t to predict when a trajectory will overlap a specific location. So, here is what I got so far.

Variable Constants:
V_i = initial velocity
a = acceleration
w = omega, angular velocity, projectiles turn rate

Variables = t , time

I started with the basic velocity equation.
v(t) = dx/dt

re-arranged it and took the integral of both sides

integral(dx) = integral(v(t)dt)

substituted v(t) for what I needed ( this equation will be the forward component )

x = integral(cos(wt)*(v_i+at)dt)

integrated the first half

x = (V_i*sin(wt))/w + ...

and then the second half and got the final equation
solving for c where at t = 0 x should be 0

$$x = \frac{V_{i}}{\omega}*sin(\omega*t) + \frac{a}{\omega^{2}}*cos(\omega*t) + \frac{a*t}{\omega}*sin(\omega*t) - \frac{a}{\omega^{2}}$$

I followed the same process for the sideways version of the equation and obtained this one.

$$y = \frac{-V_{i}}{\omega}*cos(\omega*t) + \frac{a}{\omega^{2}}*sin(\omega*t) + \frac{-a*t}{\omega}*cos(\omega*t) - \frac{V_{i}}{\omega}$$

I tested these equations and they work, much to my surprise. Tested them on a TI83 Plus using the X and Y mode. However, for my purposes I need to be able to get these in "t =" form.

I've been rearranging them back and forth for hours but I just can't seem to get it. I'm not sure it's even possible since there are several t's per x and some x's may not have a t at all so it's not exactly a function curve. Any help would be greatly appreciated.

 

[lippyka]

Thank you!To solve for t, you will need to use some trigonometric identities and algebraic manipulation to manipulate the equations into a form where you can solve for t. First, combine the x and y equations into one equation that includes both x and y:x^2 + y^2 = (V_i/w)^2 + (a/w^2)^2 + (a/w)^2*t^2 - 2*(V_i/w)*(a/w^2)*cos(wt) - 2*(a/w)*t*sin(wt) + 2*(V_i/w)*(a/w)*t*cos(wt) + 2*(V_i/w)*(a/w^2)*sin(wt).Now, you can use the identity cos^2(wt) + sin^2(wt) = 1 to simplify the equation:x^2 + y^2 = (V_i/w)^2 + (a/w^2)^2 + (a/w)^2*t^2 - 2*(V_i/w)*(a/w^2) - 2*(a/w)*t + 2*(V_i/w)*(a/w)*t.Now, you can rearrange the equation to get it in the form of a quadratic equation:(a/w)^2*t^2 - 2*(V_i/w)*(a/w)*t + (V_i/w)^2 + (a/w^2)^2 - (x^2 + y^2) = 0.Now, you can use the quadratic formula to solve for t:t = \frac{2*(V_i/w)*(a/w)*\pm\sqrt{(2*(V_i/w)*(a/w))^2 - 4*(a/w)^2*((V_i/w)^2 + (a/w^2)^2 - (x^2 + y^2))}}{2*(a/w)