A position-time ($x$–$t$) graph plots where an object is at each moment. Its most useful feature is its slope, which you can find with The Slope Formula. Over any straight piece, rise over run is $\dfrac{\Delta x}{\Delta t}$, which is exactly the Average Velocity.
Key idea
The slope of an $x$–$t$ graph is the velocity. Steeper means faster; upward slope means moving in the $+x$ direction; downward slope means $-x$; flat means standing still.
Why it works
The rise of the graph is a change in position, $\Delta x$, and the run is a change in time, $\Delta t$. Dividing gives $\Delta x / \Delta t$, the definition of velocity. The height of the graph only tells you where the object is, not how fast it moves.
| On the x–t graph | The motion |
|---|---|
| straight line sloping up | steady velocity in the +x direction |
| steeper line | faster |
| flat line | at rest |
| line sloping down | steady velocity in the −x direction |
| crossing the t axis | passing the origin (not stopping) |
Worked examples
Worked example: Reading a slope
On the graph above, find the velocity from $t = 5\ \text{s}$ to $t = 8\ \text{s}$.
- $x$ goes from $9\ \text{m}$ to $0\ \text{m}$: $\Delta x = -9\ \text{m}$.
- $\Delta t = 3\ \text{s}$, so $v = \dfrac{-9}{3} = -3\ \text{m/s}$.
Answer: $-3\ \text{m/s}$ (walking back toward the origin)
Worked example: Fastest is steepest
On the same graph, when is the walker fastest?
- Slopes: $0$–$2\ \text{s}$: $2\ \text{m/s}$; $2$–$4\ \text{s}$: $0$; $4$–$5\ \text{s}$: $4\ \text{m/s}$; $5$–$8\ \text{s}$: $-3\ \text{m/s}$.
- Speed is the size of the slope, so the biggest is $4\ \text{m/s}$, from $4$ to $5\ \text{s}$.
Answer: From $t = 4\ \text{s}$ to $5\ \text{s}$
Common mistake
Reading a high point on the graph as high speed. The top of the graph above ($x = 9\ \text{m}$ at $t = 5\ \text{s}$) is where the walker is farthest away, not where they move fastest; at the highest point of a curved graph, the object is momentarily stopped. Check: always ask "how steep?", never "how high?", when you want velocity.