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Relative to an origin, the inertia tensor is $I_{ij}=\int (r^2{\delta }_{ij}-r_i{r}_j)dm$, with $\mathbf{J}=\mathsf{I}\mathbf{\omega }$. Its orthogonal eigenvectors are principal axes, with eigenvalues the principal moments of inertia. Different components of $\mathbf{\omega }$ are multiplied by different eigenvalues, so angular momentum need not point along angular velocity.

For a uniform thin disc, radial integration gives $I_z=\int r^{2}dm=m{a}^2/2$. The perpendicular axis theorem and symmetry then give $I_x=I_y=I_z/2$, henceFor the elastic eccentric impact of a thin disc, neglect gravity's impulse and take the pole's impulse as vertical. Let $z$ point up and let the contact position relative to the centre be $r\hat{x}$. An upward impulse $P$ givesFor an elastic impact with a fixed pole, the contact point reverses its normal velocity, so $-u+P/m+r^{2}P/I_d=+u$. ThereforeHere $v_C>0$ denotes upward motion; the centre can still move down if $r>a/2$. Substitution givesThis also follows by combining conserved angular momentum about the fixed contact with conserved kinetic energy and selecting the separating, nonzero-impulse solution. At $r=0$ the disc simply rebounds without rotation; at $r=a/2$ its centre is instantaneously stationary. The rebound result concerns the contacting material point immediately after this ideal impact, not its subsequent contact history.

For an elastic eccentric impact of a thin disc, take upward velocity as positive and let $J$ be the short upward impulse. The disc's moment of inertia about a horizontal diameter is $I=M{a}^2/4$. With impact offset $d=a/2$,The contact point has upward velocity $v_C+d\omega$. Unit restitution against the fixed post requires that velocity to be $+u$, soSince $I=M{d}^2$, $J=Mu$ and the centre's instantaneous speed is zero. The disc rotates with $\omega =2u/a$; its rotational kinetic energy $I{\omega }^2/2=M{u}^2/2$ equals the entire initial energy. Gravity's impulse during the impact is neglected. Using the polar-axis moment of inertia would describe the wrong rotation axis.