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!
What is a Graph
Watch this Explanation
Time for Practice
Use the Moving Man Applet to assist in this Practice.
- The position graph below is of the movement of a fast turtle who can turn on a dime. a. Sketch the velocity vs. time graph of the turtle below. b. Explain what the turtle is doing (including both speed and direction) from: i) 0-2s. ii) 2-3s. iii) 3-4s. c. How much distance has the turtle covered after 4s?\begin{align*}{\;}\end{align*} d. What is the turtle’s displacement after 4s? \begin{align*}{\;}\end{align*}
- Draw the position vs. time graph that corresponds to the velocity vs. time graph below. You may assume a starting position \begin{align*}x_0 = 0\end{align*}. Label the \begin{align*}y-\end{align*}axis with appropriate values.
- The following velocity-time graph represents 10 seconds of actress Halle Berry’s drive to work (it’s a rough morning).
a. Fill in the tables below – remember that displacement and position are not the same thing!
Instantaneous Time (s) | Position (m) | |||
---|---|---|---|---|
Interval (s) | Displacement (m) | Acceleration\begin{align*}(m/s^2)\end{align*} | 0 sec | 0 m |
0-2 sec | ||||
2 sec | ||||
2-4 sec | ||||
4 sec | ||||
4-5 sec | ||||
5 sec | ||||
5-9 sec | ||||
9 sec | ||||
9-10 sec | ||||
10 sec |
b. On the axes below, draw an acceleration-time graph for the car trip. Include numbers on your acceleration axis.
c. On the axes below, draw a position-time graph for the car trip. Include numbers on your position axis. Be sure to note that some sections of this graph are linear and some curve – why?
- Two cars are drag racing down El Camino. At time \begin{align*}t = 0\end{align*}, the yellow Maserati starts from rest and accelerates at \begin{align*}10 \ m/s^2\end{align*}. As it starts to move it’s passed by a ’63 Chevy Nova (cherry red) traveling at a constant velocity of 30 m/s. a. On the axes below, show a line for each car representing its speed as a function of time. Label each line. b. At what time will the two cars have the same speed (use your graph)? \begin{align*}{\;}\end{align*} c. On the axes below, draw a line (or curve) for each car representing its position as a function of time. Label each curve. d. At what time would the two cars meet (other than at the start)? \begin{align*}{\;}\end{align*}
Answers:
1c. 25 m
1d. -5 m
2. discuss in class
3. discuss in class
4b. 3 sec
4d. 6 sec