Spin-resolved oblique impact dynamics: A unified framework connecting inclined-plane geometry to tangential restitution

In this study, a comprehensive experimental and theoretical investigation of oblique impact dynamics in the inclined-plane geometry is presented, incorporating simultaneous range and ball-spin measurements via 240 fps slow-motion video for the first time. A steel ball dropped from heights h = 10–50 cm onto smooth inclined planes at θ = 45° and 60° (where θ is the angle of the surface to the horizontal) yields two independent observables: the horizontal range R1 and the post-impact angular velocity ω2. It is shown that these two observables obey orthogonal angular scaling laws—range scales as sin 2θ and spin scales as cos θ (in the sliding regime)—providing a self-consistent protocol that simultaneously extracts the true normal coefficient of restitution e⊥ and the dynamic friction coefficient μd from a single inclined-plane experiment without requiring a vertical-drop calibration or a force plate. The extracted values e⊥ = 0.732 ± 0.007 and μd = 0.065 ± 0.003 are mutually consistent and independently confirmed by a separate vertical-bounce measurement. A complete energy tripartition—normal deformation, tangential sliding, and rotational kinetic energy—is derived, extending the framework of Cross (2023) to the inclined-plane geometry. A corrected elastic baseline (the horizontal range a perfectly elastic, frictionless ball would achieve under the same conditions) for θ ≠ 45° is derived and shown to eliminate a systematic error of up to 55% in the apparent coefficient of restitution at 60° introduced by the widely used θ = 45° formula. All measurements are recovered within 3% by the full spin-friction model.