Solve an off-center collision with angular impulse

Let a short collision deliver impulse J at a point whose center-of-mass offset is r. Linear momentum changes by J, while angular momentum about the center changes by r×J if other angular impulses are negligible. During an ideal instantaneous impact, position and orientation do not jump; velocities can. In three dimensions the inertia tensor in the chosen inertial coordinates must be used to convert angular impulse into an angular-velocity change. Replacing the tensor with a scalar is justified only for the appropriate planar or principal-axis problem.

For a planar rod, let a denote the horizontal contact offset and let a positive normal impulse J point upward. Then center velocity changes by J/m in the vertical direction and angular velocity changes by aJ/I about the perpendicular axis. A finite force such as gravity has negligible impulse only when its magnitude times the short impact duration is small compared with the collision impulse. This is an approximation with a time-scale justification, not a statement that gravity stops acting during contact.

For a stationary frictionless stop, define u⁻ as the incoming normal velocity of the rod contact point, negative when approaching. Its value is Vᵧ⁻ + aω⁻ for the geometry above. The outgoing value is u⁺ = Vᵧ⁺ + aω⁺. Newton restitution imposes u⁺ = −eu⁻ with 0 ≤ e ≤ 1 in this simple passive model. Solving with the impulse balances gives an effective inverse mass 1/m + a²/I. The larger the rotational response, the less impulse is needed to change the contact velocity.

The restitution law must not be applied to the center velocity unless the contact happens to move with that center in the normal direction. For two moving bodies, both contact velocities and both inverse-inertia contributions enter. A rough contact also needs tangential impulse and a friction rule; simultaneous contacts require a consistent coupled treatment. Real impacts store and dissipate energy through deformation, sound, and heat. A rigid instantaneous restitution model summarizes some of those effects and cannot predict its own coefficient e from the rigid-body assumptions alone.

The contact impulse changes angular momentum about the center unless its line of action passes through that center. About a fixed inertial point at the contact location, however, that same impulse has zero moment. Total angular momentum about that fixed point can be conserved during the short impact if all other angular impulses are negligible. The total includes both the center-of-mass orbital term and the spin term. Keeping only Iω about the contact point for a freely translating body can discard an essential piece of momentum.

Use the vector lab to resolve a planar impulse into components. Set magnitude to two and predict its projections at ninety degrees, then rotate the angle while holding the magnitude fixed. With a horizontal lever arm, only the vertical component contributes to the perpendicular angular impulse. This is a geometric check on r×J, not a collision solver: the lab displays neither a lever arm nor restitution. The actual outgoing center and contact velocities must be calculated from the impulse equations and checked for separation.