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Impulse, momentum, and kinetic-energy laws for an isolated frictionless sphere impact

Statement

Let two spheres of radii $r_{1}, r_{2} > 0$ and masses $m_{1}, m_{2} > 0$ have centers $q_{1}, q_{2} \in R^{d}$ at contact, so $∥ q_{1} - q_{2} ∥_{2} = r_{1} + r_{2}$ . Let $n = (q_{1} - q_{2}) / (r_{1} + r_{2})$ be the unit normal pointing from center $2$ to center $1$ . Suppose the pre-impact relative normal velocity is $v_{n} = ⟨ v_{1}^{-} - v_{2}^{-}, n ⟩ < 0$ . For prescribed restitution $e \in [0, 1]$ , put $α = \frac{1}{m_{1}} + \frac{1}{m_{2}}, J = - \frac{(1 + e) v_{n}}{α}, v_{1}^{+} = v_{1}^{-} + \frac{J}{m_{1}} n, v_{2}^{+} = v_{2}^{-} - \frac{J}{m_{2}} n .$ Then $J > 0$ , total linear momentum is conserved, the post-impact relative normal velocity is $- e v_{n}$ , each tangential velocity is unchanged, and $K^{+} - K^{-} = - \frac{(1 - e^{2}) v_{n}^{2}}{2 (1 / m_{1} + 1 / m_{2})} \leq 0,$ where $K^{\pm} = \frac{1}{2} m_{1} ∥ v_{1}^{\pm} ∥_{2}^{2} + \frac{1}{2} m_{2} ∥ v_{2}^{\pm} ∥_{2}^{2}$ . This is a single frictionless binary impulse law; it asserts no existence or uniqueness for simultaneous or frictional multi-contact.

Facts & Assumptions

Given: The two masses, contact normal, pre-impact velocities, and restitution coefficient in the Statement.

[F1]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : the Euclidean inner product is bilinear and symmetric, and $∥ x ∥_{2}^{2} = ⟨ x, x ⟩$ .

Proof

1.1F1givenalgebra

Since $m_{1}, m_{2} > 0$ , $α > 0$ . Since $1 + e > 0$ and $v_{n} < 0$ , $J > 0$ . The impulse changes total momentum by $m_{1} (v_{1}^{+} - v_{1}^{-}) + m_{2} (v_{2}^{+} - v_{2}^{-}) = J n - J n = 0$ .

2.1F1step 1.1givenalgebra

Taking the relative normal component gives $⟨ v_{1}^{+} - v_{2}^{+}, n ⟩ = v_{n} + J (\frac{1}{m_{1}} + \frac{1}{m_{2}}) = v_{n} - (1 + e) v_{n} = - e v_{n} .$ For either velocity, projection onto any vector orthogonal to $n$ is unchanged because the impulse term is parallel to $n$ .

3.1F1step 1.1givenalgebra∎

Expanding the two squared norms with [F1] yields $K^{+} - K^{-} = J ⟨ v_{1}^{-} - v_{2}^{-}, n ⟩ + \frac{J^{2}}{2} (\frac{1}{m_{1}} + \frac{1}{m_{2}}) = J v_{n} + \frac{J^{2} α}{2} .$ Substitute $J = - (1 + e) v_{n} / α$ to obtain $K^{+} - K^{-} = - (1 - e^{2}) v_{n}^{2} / (2 α)$ . Since $0 \leq e \leq 1$ , this is nonpositive; at $e = 0$ it is the maximal loss in this law and at $e = 1$ it is zero.

Depends on

The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$

Used by

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Sources

David E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints (standard reference, not scraped)
Dimitri P. Bertsekas, 6.253 Convex Analysis and Optimization, Complete Lecture Notes (standard reference, not scraped)