I have a situation of two 2d objects (balls with radii of 5 each) moving along straight lines. I know their velocity, their acceleration, initial coordinates and direction.
Please note, I'm not 100% sure that it is possible to answer this at all. This is not a homework and not an exercise from a textbook. So if anyone thinks that there is no way to solve this, please let me know.
If they were dots, I would know how to find the answer: simply find their trajectories intersection, distances from their initial positions to the point of the intersection and solve the equation with a time variable:
$$ V_{i1} * t + \frac{1}{2} a_1 t^2 = V_{i2} * t + \frac{1}{2} a_2 t^2 $$
That would answer the question if and when would the objects collide.
But since I have balls this approach doesn't help. They can collide much sooner then the intersection of their centers would occur.
The only thing I was able to come up with is to find an inequality of distance between the balls' centers and try to find when the distance will become less than two times a ball's radius:
$$ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} <= 10 \\ \sqrt{(x_2 - x_1)^2 + (kx_2 + b_2 - kx_1 - b_1)^2} <= 10 $$
Trying to solve the equation
$$ \sqrt{(x_2 - x_1)^2 + (kx_2 + b_2 - kx_1 - b_1)^2} = 10\\ (x_2 - x_1)^2 + (kx_2 + b_2 - kx_1 - b_1)^2 = 100 $$
to find values of $x_1$ and $x_2$ is hopeless, since there are two variables $x_1$ and $x_2$ and only one equation.
So I have no idea, where should I proceed now. Could anyone either tell me how to solve this exactly or at least suggest in what direction should I proceed?
Thank you.
