To find the proper distance between two particles in relativity, one must consider the effects of length contraction. Given the particles move at a speed of v = 3/4c and have a time interval of t = 50 ns before hitting a target, the laboratory measures their distance as vt. The proper distance, or rest length, can be calculated using the length contraction formula, which relates the measured length to the proper length. Understanding these concepts is crucial for solving such problems effectively. The discussion emphasizes the importance of grasping the foundational equations in relativity to approach these exercises confidently.