Velocity–time graphs convert motion into a visual model you can read quantitatively.

Velocity–time graphs encode two key kinematics ideas at once: the slope gives acceleration and the area under the curve gives displacement. The figure’s straight-line segments make it easy to compute area by simple geometry (rectangle + triangle), reinforcing why $\Delta x$ is found from signed area on a $v$–$t$ graph. Source

In AP Physics 1, you must connect graph features to physical meaning using slope for acceleration and area for displacement, including signs.

Velocity–Time Graph Essentials

A velocity–time (v–t) graph plots velocity on the vertical axis and time on the horizontal axis. Because velocity is a vector in one dimension, points above or below $v=0$ indicate direction.

Reading direction and motion

• $v>0$: motion in the chosen positive direction

• $v<0$: motion in the negative direction

• Crossing $v=0$: the object changes direction (it is momentarily at rest at the crossing if the graph is continuous)

Slope of a v–t Graph: Instantaneous Acceleration

The slope tells how quickly velocity changes with time. On a curved graph, the slope depends on the instant you choose, so you use a tangent line.

This is exactly the syllabus idea: instantaneous acceleration is the slope of a tangent line on a velocity–time graph. A positive slope means acceleration in the positive direction; a negative slope means acceleration in the negative direction.

A steep slope (large magnitude) means velocity is changing quickly. A horizontal segment has zero slope, so $a=0$ during that interval.

Tangent vs. secant (why it matters)

• Tangent slope: acceleration at one instant (instantaneous)

• Secant slope: average acceleration over a time interval (connect two points with a straight line)

When the graph is a straight line, tangent and secant slopes match everywhere, indicating constant acceleration.

Area Under a v–t Graph: Displacement

The second syllabus requirement is that displacement is the area under a velocity–time graph. The key is that velocity multiplied by time gives a length, so graphical area corresponds to position change.

Area is signed:

• Area above the time axis ($v>0$) contributes positive displacement.

• Area below the time axis ($v<0$) contributes negative displacement.

• Net displacement equals (positive area) minus (magnitude of negative area).

In practice, for piecewise straight graphs you typically find area by geometry:

• Rectangle area corresponds to constant velocity.

• Triangle or trapezoid areas correspond to linearly changing velocity. Be careful to include negative areas when the graph is below the axis.

What area does (and does not) tell you

• The net area gives net displacement (where you end relative to where you started).

• If you need total distance travelled, you would add the magnitudes of the areas above and below the axis (treat negative regions as positive). This is not the same as displacement when direction reverses.

Common Graph Interpretation Pitfalls

• Confusing slope with height: height is velocity; slope is acceleration.

• Forgetting that “area under the curve” is measured to the time axis, not to an arbitrary reference line.

• Ignoring sign: negative velocity implies negative area, which subtracts from net displacement.

• Using endpoints only on a curved graph: for instantaneous acceleration, you need the tangent at the time of interest.