Velocity

Remarks

Example: for uniform circular motion $x(t) = (R\cos(\omega t), R\sin(\omega t), 0)$ with $R > 0$ and $\omega \in \R$, the velocity is $v(t) = (-R\omega\sin(\omega t), R\omega\cos(\omega t), 0)$ and the speed is $R\abs{\omega}$, constant although the velocity turns. Non-example: the average velocity $(x(t_2) - x(t_1))/(t_2 - t_1)$ over an interval is not the velocity at an instant. For the circular motion over one full period it is $0$, while the speed is $R\abs\omega$. Common notations are $v$, $\dot x$ and $dx/dt$. Velocity depends on the frame of reference: in a frame moving with constant velocity $u$, the velocity of the particle is $v - u$.