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Trajectory

The trajectory of a moving object is the path it follows through space, often represented as a curve.
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The statement of the theorem

Let I=[t0,tf]⊂RI = [t_0, t_f] \subset \mathbb{R} be the time interval. Define the position vector r:I→R3\mathbf{r}: I \to \mathbb{R}^3 such that r(t)=⟨x(t),y(t),z(t)⟩\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle, where x(t),y(t),z(t)x(t), y(t), z(t) are continuously differentiable functions of tt. The trajectory T\mathcal{T} is the image set of this mapping: T=r(I)={r(t)∣t∈I}.\mathcal{T} = \mathbf{r}(I) = \{\mathbf{r}(t) \mid t \in I\}. Furthermore, the velocity vector v(t)\mathbf{v}(t) and acceleration vector a(t)\mathbf{a}(t) are defined by the derivatives: v(t)=drdt=r′(t)\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = \mathbf{r}'(t) and a(t)=d2rdt2=r′′(t).\mathbf{a}(t) = \frac{d^2\mathbf{r}}{dt^2} = \mathbf{r}''(t).