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# Graphing Motion

## Use the slope of a graph to determine instantaneous velocity or acceleration for an object.

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Graphing Motion

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 x\begin{align*}x\end{align*} vs. t\begin{align*}t\end{align*} produces a parabola. The slope of the x\begin{align*}x\end{align*} vs. t\begin{align*}t\end{align*} graph equals the instantaneous velocity. The slope of a v\begin{align*}v\end{align*} vs. t\begin{align*}t\end{align*} graph equals the acceleration.

The slope of the graph v\begin{align*}v\end{align*} vs. t\begin{align*}t\end{align*} can be used to find acceleration; the area of the graph v\begin{align*}v\end{align*} vs. t\begin{align*}t\end{align*} can be used to find displacement. Welcome to calculus!

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

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

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

### Review

1. 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*}
2. Draw the position vs. time graph that corresponds to the velocity vs. time graph below. You may assume a starting position x0=0\begin{align*}x_0 = 0\end{align*}. Label the y\begin{align*}y-\end{align*}axis with appropriate values.
3. 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(m/s2)\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?

1. Two cars are drag racing down El Camino. At time t=0\begin{align*}t = 0\end{align*}, the yellow Maserati starts from rest and accelerates at 10 m/s2\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*}

1 b. i) 10 m/s  ii) 0 m/s  iii) -15 m/s c. 25 m d. -5 m

4 b. 3 seconds   d. 6 seconds

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