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Inelastic and elastic endpoints of a scalar impact step

Example

Use $q = 1$ , $v = - 2$ , $a = 0$ , $h = 1$ , and $m = 2$ . Compare restitution values $e = 0$ and $e = 1$ , with a root-search budget of two candidate roots.

Facts & Assumptions

Given: The scalar impact inputs in the Example.

[F1]

First-impact step for a one-dimensional unilateral contact: the earliest root with incoming velocity is returned as an impact with $J = - m (1 + e) v^{-}$ and $v^{+} = - e v^{-}$ .

[F2]

Complementarity, momentum balance, and energy loss at a scalar impact: a resolved impact has nonnegative impulse, zero contact gap, momentum balance, and kinetic-energy ratio $e^{2}$ across the collision.

Verification

1.1F1givenalgebra

The free-flight gap is $g (s) = 1 - 2 s$ , whose unique root on $(0, 1]$ is $s = 1 / 2$ with $v^{-} = - 2 < 0$ . Hence the step returns an impact at contact for both restitution values.

2.1F1F2step 1.1givenalgebra∎

If $e = 0$ , then $J = 4$ and $v^{+} = 0$ ; if $e = 1$ , then $J = 8$ and $v^{+} = 2$ . In both cases $q_{*} J = 0$ and $J \geq 0$ . The pre-impact kinetic energy is $\frac{1}{2} (2) (- 2)^{2} = 4$ ; the post-impact energies are $0$ and $4$ , respectively. The momentum identities are $2 (- 2) + 4 = 0$ and $2 (- 2) + 8 = 4 = 2 (2)$ . These equal the inelastic and elastic endpoints of [F2].

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Sources

David E. Stewart, Dynamics with Inequalities: Impacts and Hard Constraints (standard reference, not scraped)