Instantaneous velocity

Gist: Average velocity taken over a shrinking interval. In the limit it becomes the derivative of position with respect to time.

$V_t={\lim }_{\Delta t\to 0}\frac{x_{t+\Delta t}-x_t}{\Delta t}=\frac{dx}{dt}$

Why it matters: This is where calculus enters kinematics. On a position-time graph it is the tangent slope at time $t$, so its sign — $>0$, $=0$, $<0$ — reads straight off the curve.

Detail: Lecture 2, p.2 and p.3

Related: Instantaneous speed, Instantaneous acceleration, Position-time graph, Physics