To find the relative velocity of approach between two particles moving towards each other, we can use the concept of relative velocity from special relativity.
Let's denote the velocity of the first particle as v1 = 0.8c (where c is the speed of light) and the velocity of the second particle as v2 = 0.5c.
The relative velocity between the two particles can be calculated using the relativistic velocity addition formula:
v_rel = (v1 + v2) / (1 + (v1 * v2) / c^2)
Substituting the given values:
v_rel = (0.8c + 0.5c) / (1 + (0.8c * 0.5c) / c^2) = (1.3c) / (1 + (0.4c^2) / c^2) = (1.3c) / (1 + 0.4) = (1.3c) / 1.4
Simplifying further:
v_rel = 0.92857c
Therefore, the relative velocity of approach between the two particles is approximately 0.92857 times the speed of light (c).