KAHIBARO
Table of Contents
Reading Force-Time Graphs
A force time graph shows how force changes during an interaction. The horizontal axis is time, and the vertical axis is force. This kind of graph is especially useful when forces are not constant, such as during a collision, a kick, or a bat hitting a ball.
The main idea is that the area under a force time graph gives the impulse delivered during the time interval. Since impulse changes momentum, the graph gives direct information about how strongly and how long an object is pushed.
The area under a force time graph equals impulse:
$$
J = \int_{t_1}^{t_2} F(t)\,dt
$$
For a constant force, this becomes:
$$
J = F \Delta t
$$
Impulse is equal to the change in momentum:
$$
J = \Delta p
$$
What the Graph Tells You
A force time graph can tell you several things at once. A tall graph means the force is large. A wide graph means the force acts for a long time. A graph with a larger area means a larger impulse.
Two different force time graphs can have the same area, and therefore the same impulse, even if one has a very large force for a short time and the other has a smaller force for a longer time. This is important in safety applications. For example, airbags and padded surfaces increase the time of impact, which reduces the average force for the same change in momentum.
Area Under the Graph
If the force is constant, the graph is a rectangle. Its area is easy to find:
$$
J = F \Delta t
$$
If the graph is triangular, the impulse is the area of the triangle:
$$
J = \frac{1}{2} \times \text{base} \times \text{height}
$$
If the graph has a more complicated shape, the impulse is found by adding the areas of simple pieces, or more generally by integration.
Common Shapes and Their Areas
| Graph shape | Area formula | Impulse |
|---|---|---|
| Rectangle | $F \Delta t$ | $J = F \Delta t$ |
| Triangle | $\frac{1}{2}bh$ | $J = \frac{1}{2}bh$ |
| Trapezoid | $\frac{1}{2}(F_1 + F_2)\Delta t$ | $J = \frac{1}{2}(F_1 + F_2)\Delta t$ |
Here, the base or width is the time interval, and the height is the force value.
Positive and Negative Force
Force can be positive or negative depending on the chosen direction. On a force time graph, parts above the time axis represent positive force, and parts below the time axis represent negative force.
This means that area above the axis counts as positive impulse, while area below the axis counts as negative impulse. If both appear, the net impulse is the signed total area.
Impulse from a graph is not always the total geometric area. It is the signed area.
Areas above the time axis are positive, and areas below are negative.
If the net area is zero, the object has no net change in momentum, even though forces may have acted during the interval.
Average Force from a Force-Time Graph
A changing force can be replaced by an average force that produces the same impulse over the same time interval. If the total impulse is known, then
$$
F_{\text{avg}} = \frac{J}{\Delta t}
$$
Using momentum,
$$
F_{\text{avg}} = \frac{\Delta p}{\Delta t}
$$
On the graph, the average force is the height of a rectangle with the same width and the same area as the actual curve.
Physical Meaning in Collisions
During a collision, the force usually rises quickly to a peak and then falls back to zero. The exact graph may be irregular, but the area still gives the impulse. A sharper, narrower peak means a large force acting briefly. A broader curve means the force is spread over more time.
For the same momentum change, increasing the collision time reduces the average force. This is why helmets, cushions, and crumple zones are effective.
For a fixed change in momentum,
$$
\Delta p = F_{\text{avg}} \Delta t
$$
If $\Delta t$ increases, then $F_{\text{avg}}$ decreases.
Example Interpretation
Suppose a force acts on an object for $0.4\,\text{s}$, reaching a maximum value of $20\,\text{N}$, and the graph is triangular. The impulse is
$$
J = \frac{1}{2}(0.4)(20) = 4\,\text{N s}
$$
Since $1\,\text{N s} = 1\,\text{kg m/s}$, the change in momentum is
$$
\Delta p = 4\,\text{kg m/s}
$$
The average force over this time is
$$
F_{\text{avg}} = \frac{4}{0.4} = 10\,\text{N}
$$
So even though the peak force is $20\,\text{N}$, the average force is only $10\,\text{N}$.
Visualizing the Graph
In this drawing, the impulse is the area of the triangle between $t_1$ and $t_2$.
Comparing Two Interactions
These two graphs can represent the same impulse if their areas are equal. The first has a larger force for a shorter time. The second has a smaller force for a longer time.
Practical Skill: How to Solve Problems
When using a force time graph, first identify the time interval where the force acts. Then determine the shape under the curve. Find the signed area. That area is the impulse. Finally, use
$$
J = \Delta p
$$
to connect the graph to the object's momentum change.
If needed, you can then find average force from the total area and total time interval.
Final Idea
Force time graphs turn a changing force into a geometric picture. The most important feature is not just the height of the graph, but the area under it. That area tells you the impulse, and impulse tells you how the momentum changes.
Key rule for force time graphs:
$$
\text{Area under the } F\text{ versus }t\text{ graph} = \text{impulse} = \Delta p
$$