Velocity Graphing Calculator

On a position-time graph the vertical axis is position $x$ in metres and the horizontal axis is time $t$ in seconds, so velocity is the slope, in m/s.

Average velocity between two points is the slope of the straight line joining them (the secant):

$v_{\text{avg}} = \frac{x_2 - x_1}{t_2 - t_1}$

Instantaneous velocity at one instant is the slope of the tangent there, which is the derivative:

$v(t) = \frac{dx}{dt}$

How to read the shape:

• Horizontal line — at rest, $v = 0$
• Straight sloped line — constant velocity; steeper means faster
• Curve bending upwards — speeding up in the positive direction
• Negative slope — moving in the negative direction

The assumption people forget: the secant slope and the tangent slope agree only when the graph is straight. On a curve, an average velocity over an interval is not the velocity at any particular instant inside it — except at one point, by the mean value theorem.

Swap the vertical axis to velocity and both features change meaning:

• Slope is acceleration, $a = \Delta v / \Delta t$, in m/s²
• Area under the curve is displacement, in metres

$\Delta x = \int_{t_1}^{t_2} v(t)\,dt$

In practice you rarely integrate. Break the shape into rectangles and triangles: a rectangle contributes $v \cdot \Delta t$, a triangle contributes $\tfrac12 \Delta v \cdot \Delta t$.

Area below the time axis counts as negative — the object is moving backwards, so it subtracts from the displacement.

When it applies: the area rule is exact for any velocity curve, straight or not; it is nothing more than integration.

The assumption people forget: the signed area gives displacement, not distance travelled. For distance, add the absolute values of the areas above and below the axis separately.

• Reading the height of a position-time graph as velocity — the height is position. Velocity is the slope.
• Mixing up the two graph types — a horizontal line means "at rest" on a position-time graph but "constant velocity" on a velocity-time graph.
• Using a tangent slope as an average — the tangent gives one instant only.
• Forgetting the sign of the area — a trip out and back has a positive area then a negative one, and they cancel to zero displacement.
• Ignoring the axis scale — a squashed vertical axis makes a gentle slope look dramatic. Always compute from the numbers, not the visual steepness.
• Assuming a curved position-time graph means changing acceleration — a parabola means constant acceleration.