To calculate the initial velocity of a metal ball launched from a ramp with a given horizontal range and launch angle, the equations of projectile motion can be utilized. The horizontal range can be expressed as a function of initial velocity, launch angle, and vertical displacement. Key equations include the range formula, which incorporates both horizontal and vertical components of motion. By rearranging these equations, the initial velocity can be isolated and calculated. Understanding the relationship between the angle, range, and displacement is crucial for accurate computation.