Foundations — Average velocity vs instantaneous velocity

Before you can read the parent note, you must be fluent in the little pieces of notation it throws at you. Below, every single symbol is earned: plain words first, then a picture, then why the topic needs it. Nothing is used before it is built.

1. What is a "position"? — the symbol $\vec{r}$

The picture. Put a dot called the origin on your page — call it $O$. The object sits somewhere else. Draw an arrow from $O$ straight to the object. That arrow is $\vec{r}$.

Why the topic needs it. Velocity is about motion, and motion means the position changes. You cannot talk about a change in position until you can name a position. $\vec{r}$ is that name.

See Position and displacement vectors for the full construction.

2. Time as a label — the symbol $t$ and $\vec{r}(t)$

The picture. Imagine a film of the moving object. Each frame is stamped with a time $t$ (in seconds). $\vec{r}(t)$ is a machine: put in a time, out comes the arrow $\vec{r}$ for that frame.

Why the topic needs it. To ask "how fast is it moving?" you compare where it is now with where it was a moment ago. That needs position pinned to two different times — exactly what $\vec{r}(t_1)$ and $\vec{r}(t_2)$ give.

3. The Greek letter $\Delta$ ("delta") — "change in"

The picture. Two arrows from the origin: the short one is $\vec{r}(t_1)$ (start), the long one is $\vec{r}(t_2)$ (end). $\Delta \vec{r}$ is the new arrow drawn from the tip of the start-arrow to the tip of the end-arrow — the shortcut from where you were to where you ended.

Why the topic needs it. This arrow $\Delta \vec{r}$ is called the displacement — the net change in position. The whole idea of velocity is "displacement divided by time," so $\Delta$ is the heart of the machinery. Contrast displacement with path length in Distance vs displacement.

4. A rate — dividing a change by the time it took

The picture — a secant line. Draw a graph with time along the bottom and position up the side (a position–time graph — see Position-time graphs). Mark the two points $(t_1,x_1)$ and $(t_2,x_2)$. The straight line joining them is the secant (a chord). Its steepness — how much it rises for each step across — is the average velocity.

Why the topic needs it. "Average velocity = slope of the secant" is one of the two headline facts of the whole parent note. You cannot see it without knowing what slope and secant mean.

5. Shrinking the interval — the limit $\lim$ and $\Delta t\to 0$

The picture. Keep the start point fixed and slide the end point of the secant closer and closer to it. The chord pivots. In the limit — when the two points touch — the secant becomes the tangent: the straight line that just grazes the curve at that one point.

6. The derivative — the symbol $\frac{d\vec{r}}{dt}$

Why the topic needs it. "Instantaneous velocity = slope of the tangent = the derivative" is the other headline fact. Once you own $\lim$, tangent, and $d/dt$, the parent note's boxed formula is just plain English.

The same limit idea, applied to velocity instead of position, gives Acceleration as derivative of velocity.

7. Two families to keep apart: vector vs scalar

The picture. Velocity is an arrow; its magnitude $|\vec{v}|$ is just the arrow's length — that length is the speed. Direction thrown away. This is why a round trip can have zero velocity (arrows cancel) but nonzero speed (lengths add up). Compare in Average speed vs instantaneous speed.

These position functions come from the Equations of motion under uniform acceleration.

Prerequisite map

Equipment checklist