Knowing how far a particle has gone is only half the story; we also want to know how fast and how its motion is changing. Average speed is total path length divided by total time, a scalar: $\text{speed}_{avg} = \dfrac{\text{path length}}{\Delta t}$. Average velocity is total displacement divided by total time, a vector that carries the sign of $\Delta x$: $\bar{v} = \dfrac{\Delta x}{\Delta t} = \dfrac{x_2 - x_1}{t_2 - t_1}$. The SI unit of both is $\text{m/s}$, and the handy road conversion is $1\,\text{km/h} = \dfrac{5}{18}\,\text{m/s}$.
Average values hide the moment-to-moment detail. To capture the speedometer reading at one instant we shrink the time interval to zero. The instantaneous velocity is the limit of average velocity as $\Delta t \to 0$, which is exactly the calculus derivative of position with respect to time: $v = \lim_{\Delta t \to 0} \dfrac{\Delta x}{\Delta t} = \dfrac{dx}{dt}$. Geometrically, $v$ is the slope of the tangent to the position-time ($x$-$t$) graph at that instant. The magnitude of instantaneous velocity equals the instantaneous speed.
When velocity itself changes, the particle accelerates. Average acceleration is the change in velocity over the time taken: $\bar{a} = \dfrac{v_2 - v_1}{t_2 - t_1} = \dfrac{\Delta v}{\Delta t}$. Pushing $\Delta t \to 0$ gives the instantaneous acceleration, the derivative of velocity: $a = \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}$. Acceleration is a vector and its SI unit is $\text{m/s}^2$. On a velocity-time ($v$-$t$) graph, $a$ is the slope of the curve.
Signs deserve care in 1D. Acceleration is positive when velocity increases in the $+x$ sense and negative when velocity decreases — but a negative acceleration does not always mean slowing down. If a particle already moves in the $-x$ direction, a negative acceleration speeds it up. The reliable rule is: the particle speeds up when $v$ and $a$ have the same sign and slows down (retardation) when they have opposite signs.
A special, very common case is uniform acceleration, where $a$ is constant in both magnitude and direction. Free fall near the Earth's surface (ignoring air) is the classic example, with $a = g \approx 9.8\,\text{m/s}^2$ directed downward. Under uniform acceleration the velocity changes linearly with time, which is what makes the equations of motion in the next topic so simple.
- Average velocity $\bar{v} = \dfrac{\Delta x}{\Delta t}$ (vector); average speed uses path length (scalar).
- Instantaneous velocity $v = \dfrac{dx}{dt}$ = slope of the $x$-$t$ tangent.
- Acceleration $a = \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}$ = slope of the $v$-$t$ graph, unit $\text{m/s}^2$.
- Speeds up when $v$ and $a$ share a sign; slows down (retardation) when they differ.
- Uniform acceleration: $a$ constant; velocity changes linearly with time.