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Mechanical energy is the sum of kinetic energy and potential energy. It is conserved when all forces are conservative and the potential has no explicit time dependence.

A uniform solid sphere of mass $M$ and radius $a$ has the indicated moment of inertia about any diameter. Integrating the squared perpendicular distance against uniform mass density gives $I=8\pi \rho a^5/15=2M{a}^2/5$. It is different from the value for a thin spherical shell and determines the rotational contribution to kinetic energy during rolling without slipping.

For equal-mass elastic collisions, one reflection parametrization is $v^{\prime }=v-((v-v_{\ast })\cdot n)n$ and $v_{\ast }^{\prime }=v_{\ast }+((v-v_{\ast })\cdot n)n$. The fixed-$n$ transformation preserves pairwise momentum, kinetic energy and velocity-pair measure. A nonnegative kernel invariant under particle interchange and collision reversal yields a gain-loss operator. Its weak Boltzmann collision identity proves conservation and its entropy identity proves the Boltzmann H theorem. Unlike the Maxwell molecule collision operator, this general kernel may depend on relative speed.

Integration by parts against $1$, $v_i$ and $|v{|}^2/2$ gives these equations for mass, unnormalized momentum and kinetic energy. With $M_0\ne 0$, the mean is $u(t)=e^{-t}{u}_0$, and $E(t)=d{M}_0/2+(E_0-d{M}_0/2)e^{-2t}$. Normalizing by mass is invalid if that mass is zero.