Condition for two points to collide with constant speed $v_1$ and $v_2$ at any point of time? We have been given the initial speed of 2 particles. They have a constant speed. At any point of time $t\geq0$, their positions are given by $i+v\cdot t$ where $i$ is the initial position of the particle. So what is the condition that these 2 particles will collide?
Example: let's say position of the first particle is 1 and its speed is 3 and the position of the second particle is 3 and its speed is 1. Then they will collide at $t=1$. So how we will be sure that two points will collide if these parameters are given ?
Note: Consider their masses to be negligible
 A: Here's one idea: The trajectory of particle $i$ can be represented in parametric form: $\vec{p_i(t)}=(x_i(t), y_i(t), z_i(t))$
I'm going to assume that when you said, "constant speed," you actually meant constant velocity. If that's true, then the $x_i$, $y_i$, and $z_i$ functions all will be linear functions.
So does the line $x_1(t)$ intersect the line $x_2(t)$? If not, then there is no collision. But, if the lines do intersect, then how about the lines $y_1$ and $y_2$? Do they intersect? And, do they intersect at the same $t$? Finally, how about $z_1$ and $z_2$?
If all three pairs of 2D lines intersect, and if all three of them intersect at the same $t$, then there must exist some point in 4D spacetime that is on both trajectories (i.e., the particles collide.)
A: A collision means that the two particles exist at the same point in space at the same instant in time. Therefore, if the motion of particle 1 and particle 2 is given by $x_1(t)$ and $x_2(t)$ respectively, then a collision will occur if there exists a time $T\geq0$ such that $x_1(T)=x_2(T)$. Note that this is true for any two functions $x_1(t)$ and $x_2(t)$, even if the velocity is changing or if the motion is in more than one dimension.
A: If at least one of the particles has a finite size and both have velocity vectors directed toward the center of mass (in the center of mass system), then they will collide.
