Students will learn how to graph motion vs time. Specifically students will learn how to take the slope of a graph and relate that to the instantaneous velocity or acceleration for position or velocity graphs, respectively. Finally students will learn how to take the area of a velocity vs time graph in order to calculate the displacement.

### Key Equations

For a graph of position vs. time. The slope is the rise over the run, where the rise is the displacement and the run is the time. thus,

Slope = \begin{align*}v_{avg} = \frac{\Delta x}{\Delta t}\end{align*}

**Note**: Slope of the tangent line for a particular point in time = the instantaneous velocity

For a graph of velocity vs. time. The slope is the rise over the run, where the rise is the change in velocity and the run is the time. thus,

Slope = \begin{align*}a_{avg} = \frac{\Delta v}{\Delta t}\end{align*}

**Note**: Slope of the tangent line for a particular point in time = the instantaneous acceleration

- One must first read a graph correctly. For example on a position vs. time graph (thus the position is graphed on the y-axis and the time on the x-axis) for a given a data point, go straight down from it to get the time and straight across to get the position.
- If there is constant acceleration the graph \begin{align*}x\end{align*} vs. \begin{align*}t\end{align*} produces a parabola. The slope of the \begin{align*}x\end{align*} vs. \begin{align*}t\end{align*} graph equals the instantaneous velocity. The slope of a \begin{align*}v\end{align*} vs. \begin{align*}t\end{align*} graph equals the acceleration.
- The
**slope**of the graph \begin{align*}v\end{align*} vs. \begin{align*}t\end{align*} can be used to find**acceleration;**the**area**of the graph \begin{align*}v\end{align*} vs. \begin{align*}t\end{align*} can be used to find**displacement.**Welcome to calculus!