KAHIBARO
Table of Contents
Reading velocity, acceleration, and displacement from a velocity-time graph
A velocity-time graph shows how velocity changes with time. The horizontal axis represents time, and the vertical axis represents velocity. This graph is especially useful in motion with constant acceleration because the graph becomes very simple and easy to interpret.
If the motion is in one dimension, positive velocity means motion in the chosen positive direction, and negative velocity means motion in the opposite direction. A point on the graph tells you the velocity at that exact time.
The shape of the graph for constant acceleration
When acceleration is constant, velocity changes by equal amounts in equal time intervals. This means the velocity-time graph is a straight line.
If the object is speeding up in the positive direction, the line slopes upward. If the object is slowing down while moving in the positive direction, the line slopes downward. If the line crosses the time axis, the velocity changes sign, which means the object reverses direction.
For motion with constant acceleration, the velocity-time graph is a straight line.
Its equation is
$$
v = v_0 + at
$$
where $v_0$ is the initial velocity and $a$ is the constant acceleration.
Slope of a velocity-time graph
The slope of a velocity-time graph gives the acceleration. Slope tells us how much velocity changes per unit time.
For two points on the graph, the slope is
$$
\text{slope} = \frac{\Delta v}{\Delta t}
$$
Since acceleration is the rate of change of velocity,
$$
a = \frac{\Delta v}{\Delta t}
$$
If the graph is a straight line, the slope is the same everywhere, so the acceleration is constant.
The slope of a velocity-time graph is acceleration:
$$
a = \frac{\Delta v}{\Delta t}
$$
Positive slope means positive acceleration.
Negative slope means negative acceleration.
Zero slope means zero acceleration.
Area under the graph
The area between the graph and the time axis gives displacement. This is one of the most important ideas in a velocity-time graph.
If the graph stays above the time axis, the area is positive. If it stays below the axis, the area is negative. If the graph crosses the axis, the total displacement is the algebraic sum of the positive and negative areas.
For constant acceleration, the area often has the shape of a rectangle, a triangle, or a trapezoid.
The signed area under a velocity-time graph gives displacement:
$$
\Delta x = \text{area under the } v\text{-}t \text{ graph}
$$
Do not confuse displacement with distance. Distance uses total area without subtracting negative parts.
Common graph patterns
A horizontal line means the velocity is constant. Since the slope is zero, the acceleration is zero.
An upward sloping line means velocity is increasing with time. The acceleration is positive.
A downward sloping line means velocity is decreasing with time. The acceleration is negative.
A line below the time axis means the object is moving in the negative direction.
A line that crosses the time axis shows that the object stops for an instant and then moves in the opposite direction.
Finding displacement from simple areas
Suppose the velocity remains constant at $v$. Then the graph is a horizontal line, and the area under it is a rectangle:
$$
\Delta x = vt
$$
Suppose velocity increases linearly from $v_0$ to $v$ over time $t$. Then the area under the graph is a trapezoid:
$$
\Delta x = \frac{v_0 + v}{2} t
$$
This is the average velocity times time.
If the initial velocity is zero, then the trapezoid becomes a triangle plus possibly a rectangle of zero height, so
$$
\Delta x = \frac{1}{2}vt
$$
for that special case.
Relation to average velocity
For constant acceleration, average velocity is the midpoint between the initial and final velocities:
$$
v_{\text{avg}} = \frac{v_0 + v}{2}
$$
Then displacement can be written as
$$
\Delta x = v_{\text{avg}} t = \frac{v_0 + v}{2} t
$$
This matches the area under the velocity-time graph.
For constant acceleration,
$$
v_{\text{avg}} = \frac{v_0 + v}{2}
$$
and
$$
\Delta x = \frac{v_0 + v}{2} t
$$
These come directly from the trapezoid area on the velocity-time graph.
Interpreting sign changes
A negative velocity does not mean the object is slowing down. It only means the object is moving in the negative direction. To know whether the object is speeding up or slowing down, compare the signs of velocity and acceleration.
If velocity and acceleration have the same sign, speed increases. If they have opposite signs, speed decreases.
For example, if $v < 0$ and the slope is also negative, the object is moving in the negative direction and speeding up. If $v < 0$ but the slope is positive, the object is moving in the negative direction and slowing down.
Example table
| Graph feature | Physical meaning |
|---|---|
| Horizontal line | Constant velocity, zero acceleration |
| Positive slope | Positive acceleration |
| Negative slope | Negative acceleration |
| Above time axis | Positive velocity |
| Below time axis | Negative velocity |
| Crossing time axis | Change of direction |
| Area under curve | Displacement |
Visual example
In this graph, the line has a positive slope, so acceleration is positive. The graph starts below the time axis, so the initial velocity is negative. The line crosses the axis, so the object stops momentarily and then moves in the positive direction.
Geometric view of displacement
The shaded region is a trapezoid. Its area is
$$
\Delta x = \frac{v_0 + v}{2} t
$$
which gives the displacement during the interval.
Frequent mistakes
One common mistake is to confuse a velocity-time graph with a position-time graph. In a velocity-time graph, slope gives acceleration, not velocity.
Another common mistake is to think that the area under the graph gives acceleration. It does not. The area gives displacement.
A third mistake is to ignore negative area. If part of the graph is below the time axis, that part contributes negative displacement.
On a velocity-time graph,
slope $\rightarrow$ acceleration,
area under the graph $\rightarrow$ displacement.
Final idea
A velocity-time graph turns motion into geometry. In constant acceleration, the graph is a straight line, the slope gives acceleration, and the area gives displacement. Once you can read these two features, slope and area, you can extract most of the important information about the motion.