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Velocity

Velocity is the rate of change of displacement with respect to time, describing both speed and direction.
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The statement of the theorem

Let r:I→Rn\mathbf{r}: I \to \mathbb{R}^n be the position vector of a particle, where I=[t0,tf]⊂RI = [t_0, t_f] \subset \mathbb{R} is the time interval. The velocity vector v(t)\mathbf{v}(t) is defined as the first-order time derivative of the position vector: v(t)=drdt=lim⁡Δt→0r(t+Δt)−r(t)Δt.\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = \lim_{\Delta t \to 0} \frac{\mathbf{r}(t + \Delta t) - \mathbf{r}(t)}{\Delta t}. If r(t)=(x(t),y(t),z(t))\mathbf{r}(t) = (x(t), y(t), z(t)), then the components are given by: v(t)=(dxdt,dydt,dzdt)=(x˙(t),y˙(t),z˙(t)).\mathbf{v}(t) = \left( \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right) = \left( \dot{x}(t), \dot{y}(t), \dot{z}(t) \right). The magnitude (speed) is then the Euclidean norm: ∣v(t)∣=(dxdt)2+(dydt)2+(dzdt)2.|\mathbf{v}(t)| = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}.