Editing Kinematics (section)

===Velocity and speed===
The [[velocity]] of a particle is a vector quantity that describes the ''direction'' as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle ([[Displacement (geometry)|displacement]]) by the time interval. This ratio is called the [[average velocity]] over that time interval and is defined as<math display="block"> \mathbf\bar v = \frac{\Delta \mathbf r}{\Delta t} = \frac{\Delta x}{\Delta t}\hat\mathbf x + \frac{\Delta y}{\Delta t}\hat\mathbf y + \frac{\Delta z}{\Delta t}\hat\mathbf z =\bar v_x\hat\mathbf x + \bar v_y\hat\mathbf y + \bar v_z \hat\mathbf z \,</math>where <math>\Delta \mathbf{r}</math> is the displacement vector during the time interval <math>\Delta t</math>. In the limit that the time interval <math>\Delta t</math> approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector,
<math display="block"> \mathbf v
= \lim_{\Delta t\to 0}\frac{\Delta\mathbf{r}}{\Delta t}
= \frac{\text{d}\mathbf r}{\text{d}t}
= v_x\hat\mathbf x + v_y\hat\mathbf y + v_z \hat\mathbf z .</math>
Thus, a particle's velocity is the time rate of change of its position. Furthermore, this velocity is [[tangent]] to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.

The [[speed]] of an object is the magnitude of its velocity. It is a scalar quantity:
<math display="block"> v=|\mathbf{v}|= \frac {\text{d}s}{\text{d}t},</math>
where <math>s</math> is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, <math>\frac{\text{d}s}{\text{d}t}</math> is non-negative, which implies that speed is also non-negative.