KAHIBARO
Table of Contents
Reading acceleration as a graph
An acceleration-time graph shows how acceleration changes with time. On the horizontal axis we place time, and on the vertical axis we place acceleration. In motion with constant acceleration, this graph is especially simple because the acceleration does not change. That means the graph is a horizontal line.
If the acceleration is positive, the line lies above the time axis. If the acceleration is negative, the line lies below the time axis. If the acceleration is zero, the line lies exactly on the time axis.
What the graph tells you
The height of the graph at any moment gives the acceleration at that moment. Because the line is horizontal in constant acceleration motion, the value is the same at every time.
For example, if the graph is a horizontal line at $a = 3 \,\text{m/s}^2$, then the object’s velocity changes by $3 \,\text{m/s}$ every second. If the graph is at $a = -2 \,\text{m/s}^2$, then the velocity changes by $-2 \,\text{m/s}$ every second.
For constant acceleration, the acceleration-time graph is a horizontal line.
If the line is above the time axis, $a>0$.
If the line is below the time axis, $a<0$.
If the line is on the time axis, $a=0$.
Area under the graph
The most important use of an acceleration-time graph is that the area under the graph gives the change in velocity.
If acceleration is constant, this area is a rectangle. Its height is $a$ and its width is the time interval $\Delta t$. So the area is
$$
\Delta v = a \Delta t
$$
This matches the usual constant-acceleration relation for velocity change.
If the acceleration is negative, the area is negative, and the velocity decreases.
The area under an acceleration-time graph equals the change in velocity:
$$
\Delta v = \int a \, dt
$$
For constant acceleration,
$$
\Delta v = a \Delta t
$$
Positive and negative areas
An area above the time axis counts as positive. An area below the time axis counts as negative. This sign matters because it tells you whether velocity is increasing or decreasing in the chosen positive direction.
Here is a simple comparison.
| Graph position | Sign of acceleration | Change in velocity |
|---|---|---|
| Above time axis | Positive | Positive |
| On time axis | Zero | Zero |
| Below time axis | Negative | Negative |
A negative acceleration does not always mean the object is slowing down in an everyday sense. It means the acceleration points in the negative direction of the chosen axis. The question of whether the object speeds up or slows down depends on both velocity and acceleration, which belongs more fully to the study of velocity and direction.
Finding velocity from the graph
If you know the initial velocity $v_0$, then after a time $t$ the velocity is
$$
v = v_0 + at
$$
This comes directly from the area under the acceleration-time graph from $0$ to $t$.
For example, suppose an object starts with $v_0 = 4 \,\text{m/s}$ and has constant acceleration $a = 2 \,\text{m/s}^2$ for $3\,\text{s}$. The area under the graph is
$$
\Delta v = a\Delta t = (2)(3) = 6 \,\text{m/s}
$$
So the final velocity is
$$
v = 4 + 6 = 10 \,\text{m/s}
$$
Comparing acceleration-time and velocity-time graphs
An acceleration-time graph does not directly show position. It shows how velocity changes. So its main connection is with the velocity-time graph.
If the acceleration-time graph is a horizontal line, then the velocity-time graph is a straight slanted line. A larger constant acceleration gives a steeper velocity-time line. A negative constant acceleration gives a downward slope on the velocity-time graph.
This relationship helps you move from one graph to another. Constant acceleration means steady change in velocity.
A simple example
Imagine a car that accelerates uniformly at $1.5 \,\text{m/s}^2$ for $8\,\text{s}$. The acceleration-time graph is a horizontal line at $1.5$ from $t=0$ to $t=8$.
The change in velocity is the area of the rectangle:
$$
\Delta v = (1.5)(8) = 12 \,\text{m/s}
$$
So over those $8$ seconds, the velocity increases by $12 \,\text{m/s}$.
If instead the car had acceleration $-1.5 \,\text{m/s}^2$ for the same time, then
$$
\Delta v = (-1.5)(8) = -12 \,\text{m/s}
$$
Its velocity would decrease by $12 \,\text{m/s}$.
Common mistakes
One common mistake is to read the value of acceleration as velocity. The vertical axis gives acceleration, not speed or velocity.
Another common mistake is to ignore the sign of the area. A region below the time axis gives a negative change in velocity.
A third common mistake is to confuse the slope of the graph with the velocity change. For an acceleration-time graph, the important quantity is usually the area under the graph. In the special case of constant acceleration, the slope of the graph is zero because the graph is horizontal.
On an acceleration-time graph, the value of the graph gives acceleration, and the area under the graph gives change in velocity.
Do not confuse this with a velocity-time graph, where area gives displacement.
Summary
For motion with constant acceleration, the acceleration-time graph is a horizontal line. Its vertical position gives the constant acceleration value. The area under the graph over a time interval gives the change in velocity. For constant acceleration, this area is a rectangle, so
$$
\Delta v = a \Delta t
$$
This makes acceleration-time graphs a clear visual tool for understanding how velocity changes steadily with time.