How to Read Acceleration from a Velocity–Time Graph

Acceleration is the slope

On a velocity–time graph, acceleration is the slope. For one-dimensional motion,

$$a_x=\frac{dv_x}{dt}.$$

A horizontal segment has zero slope. A line that rises with time has positive acceleration, and a line that falls has negative acceleration. The sign describes the chosen coordinate direction; it does not by itself say whether the object is speeding up.

Separate sign from speeding up

An object speeds up when velocity and acceleration have the same sign and slows down when they have opposite signs. If $v_x<0$ and $a_x<0$, the object moves in the negative direction with increasing speed. If $v_x>0$ and $a_x<0$, its positive velocity is decreasing.

For example, let

$$v_x(t)=12-3t\quad\mathrm{m/s}.$$

The graph is a straight line with slope $-3\,\mathrm{m/s^2}$, so acceleration is constant and negative. From $t=0$ to $4\,\mathrm{s}$, velocity falls from $12\,\mathrm{m/s}$ to zero and the object slows down. After $4\,\mathrm{s}$, velocity is negative while acceleration remains negative, so the object speeds up in the negative direction.

For a curved velocity–time graph, the slope at a point is the instantaneous acceleration. A secant line across an interval instead gives average acceleration. The area under a velocity–time graph has a different meaning: it gives displacement, not acceleration.

A flat speed graph is not always enough

Velocity is a vector. In more than one dimension, its direction can change even when its magnitude, the speed, stays fixed. Uniform circular motion is the standard example: the speed is constant, but the tangent direction changes continuously, producing inward centripetal acceleration.

Use a component velocity–time graph to read component acceleration. When a trajectory curves, also inspect the changing direction of the full velocity vector instead of relying only on a speed value.