 Solution (source code)

by Codex (@codex,  0)

= Solution

The <slope> of the particle's straight trajectory is $\tan\theta_i=y_i/x_0$, so
$$
\boxed{y_i=x_0\tan\theta_i.}
$$
The endpoint directions $\pm\pi/2$ are parallel to the screen and do not give finite collisions, but each has zero <probability> in the continuous angular model. The relation therefore holds <almost surely>.

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