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Special Trajectory Equation - Solving for T

  1. Sep 14, 2007 #1
    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

    [tex]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}}[/tex]

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

    [tex]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}[/tex]

    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.
  2. jcsd
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