{"knowledge_schema": {"broad_category": "Physics \u2192 Relativity \u2192 Particle Physics", "refinement": "This problem involves understanding the behavior of particles traveling at relativistic speeds and how their decay affects their detection in a spherical detector.", "specific_scope": "The focus is on determining the required Lorentz factor for a specific proportion of particles to reach the detector's inner walls, given their decay characteristics and the geometry of the detector.", "goal": "Calculate the Lorentz factor needed for approximately two thirds of the fast-decaying particles to reach the inner walls of the spherical detector."}, "summary": "This problem is rooted in the principles of relativistic physics, specifically concerning the behavior of particles at high velocities. It requires understanding how the Lorentz factor influences the distance traveled by particles before they decay. The task is to determine the Lorentz factor necessary for two thirds of the particles to reach the inner walls of a spherical detector with a radius of 30 meters, based on their decay properties. The solution involves analyzing the relationship between the Lorentz factor, decay time, and distance traveled."}