This equation is for the x displacement of the red particle.
This equation is for the x displacement of the blue particle. Since the initial x co-ordinates of the two are same, the difference between the two x displacements would be equal to the x distance between the two.
This x distance is shown as a white line in the animation. Choosing the line joining the two balls as a reference and splitting the velocity of the red ball in to components( notice that the angle between the red ball's velocity and line joining the two balls is same as the angle between blue ball's velocity and the x-axis), the equation for the change in the length of the line joining the two balls is
This distance is shown as a yellow line in the animation. The green line gives the y-distance between the two balls. The equation for this is not necessary for the analysis. In case when the velocities are equal the right hand sides of the last two equations add up to zero. This would mean that the sum of the changes in the x-distance represented by the length of the white line and the distance- represented by the yellow line must be zero. In the animation the increase; in x-distance and; the decrease in the distance between the particles would be same. Since initially the x-distance is zero this sum is simply equal to the initial distance, which is 250 units. After a long time the x- distance between the particles should approach the distance between the particles as the y distance tends to zero. And since finally ( at t= infinity) the x-distance and distance should be equal ( because y-distance is zero), that distance should be half of the initial distance. In this case that is 125 units. A word about the accompanying chase table animation. Here each of the moving particles is advanced a small amount assuming the initial conditions to hold good for a certain duration. At the end of this duration the direction of the velocity of the chasing particle (blue) is changed and redirected towards the new position of the chased particle (red) and so on. This could be used to illustrate the chase as a stepwise process and as the number of steps increases the chasing particle follows a smooth curve. Set the time step; and the velocities of the particles to the least values and this should become evident.