Force-Time Graphs

When examining force vs time graphs, the area under the curve in a given interval of time represents the total impulse delivered over that duration of time.

For a constant force, the graph forms a horizontal line. The area bounded by this line is a rectangle with height equal to the average force $F_{\text{avg}}$ and width equal to the time interval $\Delta t$.

For negative forces, the area is negative and therefore contributes a negative impulse to the total.

Non-Constant Forces

When the force varies continuously, such as during a collision where force ramps up and down, the graph often forms shapes like triangles or trapezoids. To calculate the total impulse, decompose the area under the curve into simple geometric regions like rectangles and triangles. Calculate the area of each section individually.

For a section where the force changes linearly from zero to a peak value $F_\text{max}$ over a duration $\Delta t$, the area is that of a triangle.

Summing these individual geometric areas yields the total impulse. Units of Newtons ($\mathrm{N}$) for force and seconds ($\mathrm{s}$) for time result in impulse units of Newton-seconds ($\mathrm{N} \cdot \mathrm{s}$).