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The Alfvén velocity is the signed vector ${\mathbf{v}}_a=\mathbf{B}/\sqrt{{\mu }_0\rho }$, whose magnitude is the Alfvén speed. It follows the orientation of the magnetic field, unlike the nonnegative scalar speed.

Angular speed is the nonnegative magnitude of angular velocity: $\omega =|\mathbf{\omega }|$. It measures the rate of rotation about an instantaneous axis without specifying its direction. For uniform circular motion of radius $R$, the ordinary speed is $v=R\omega$.

For an isotropic harmonic oscillator with inertial frequency $2\omega$, viewed in a frame rotating at $\omega$, initial rotating coordinates $(1,0)$ and zero relative velocity produce the displayed astroid trajectory. The complex coordinate satisfies $\ddot{\zeta }+2i\omega \dot{\zeta }+3{\omega }^2\zeta =0$, with solution $\zeta =\frac{3}{4}{e}^{i\omega t}+\frac{1}{4}{e}^{-3i\omega t}$. The relative speed is $(3|\omega |/2)|\sin 2\omega t|$, attaining its first maximum at $\pi /(4|\omega |)$ for nonzero $\omega$.

For unit mass, incoming speed $v>0$, impact parameter $b>0$, and potential energy $\mp \kappa /r$, put $q=\kappa /v^2>0$. At closest approach, conservation of energy and conservation of angular momentum give the displayed radii and tangential speeds $u_{\mp }=bv/p_{\mp }$. Consequently $p_{-}{p}_{+}=b^2$ and $u_{-}{u}_{+}=v^2$. These relations compare equal force magnitudes. The attractive radial case $b=0$ is a singular collision, so the speed-product statement does not extend to that endpoint.

The isotropic three-dimensional Cauchy velocity distribution $f_0={\rho }_{0}b/[{\pi }^2(v^2+b^2{)}^2]$, $b>0$, has density ${\rho }_0$ and one-dimensional marginal $F(u)={\rho }_{0}b/[\pi (u^2+b^2)]$. Its second velocity moment diverges, so $b$ is a scale rather than a finite velocity dispersion. At zero phase speed, $\int F^{\prime }(u)/udu=-{\rho }_0/b^2$, giving $k_J=\sqrt{4\pi G{\rho }_0}/b$. In the upper-half frequency plane, the marginal response is ${\rho }_0/(\omega /k+ib{)}^2$; the growing branch has $\omega =i(\sqrt{4\pi G{\rho }_0}-kb)$ for $0<k<k_J$.

A classical rocket ejecting fuel at rate $\alpha >0$ and relative exhaust speed $u>0$, while collecting stationary dust of mass $\beta dx$ per traveled distance, obeysThese are mass conservation and momentum conservation for the rocket, incoming dust and outgoing gas. The dust-loading term supplies the momentum needed to accelerate incoming stationary material. Put $\lambda =\sqrt{\alpha /(\beta u)}$ and $V=\lambda u$. While the engines operate and $0\le v<V$, an initially stationary rocket satisfiesFor $\lambda >1$, continuing to $V$ would drive total mass to zero. For $0<\lambda <1$, total mass instead grows without bound and $V$ is approached only after infinite time under indefinitely maintained thrust. In both cases a finite fuel supply runs out before that limiting speed. Total stored mass must not be confused with remaining fuel.

Linear drag opposes the velocity relative to the surrounding medium with magnitude proportional to its speed, where $\alpha >0$. Its power is ${\mathbf{F}}_d\cdot \mathbf{v}=-\alpha |\mathbf{v}{|}^2$, so it dissipates kinetic energy. Under a constant external force, the resulting linear ordinary differential equation relaxes exponentially on the time scale $m/\alpha$.

A stationary particle of rest mass $m$ absorbing a photon of energy $E$ acquires total invariant massThis follows from four-momentum conservation, since the incoming photon has momentum magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter rest mass is $M/2$, attained when both are at rest in the centre-of-momentum frame. Their common laboratory speed at threshold is $cE/(m{c}^2+E)$. Above this daughter-mass threshold there is insufficient centre-of-momentum energy; below it, relative kinetic energy is available.