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Uniform Motion

Uniform motion describes a situation where an object moves with constant velocity, meaning its displacement changes linearly with time.
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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 condition for Uniform Motion is defined by the constancy of the velocity vector v(t)=drdt\mathbf{v}(t) = \frac{d\mathbf{r}}{dt}. Mathematically, this requires that the acceleration vector a(t)\mathbf{a}(t) vanishes identically: a(t)=dvdt=d2rdt2=0\mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2} = \mathbf{0}. Integrating this second-order ordinary differential equation (ODE) yields the general solution for the position r(t)\mathbf{r}(t): \n\nr(t)=r0+v0(t−t0)\mathbf{r}(t) = \mathbf{r}_0 + \mathbf{v}_0 (t - t_0) \n\nwhere r0=r(t0)\mathbf{r}_0 = \mathbf{r}(t_0) is the initial position vector, and v0=v(t0)\mathbf{v}_0 = \mathbf{v}(t_0) is the constant initial velocity vector. This solution defines a trajectory that is a straight line parameterized linearly by time.
Source: Wikipedia