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Graphing Motion

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Graphing Motion

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 = v_{avg} = \frac{\Delta x}{\Delta t}

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 = a_{avg} = \frac{\Delta v}{\Delta t}

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

Guidance
  • 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 vs. t produces a parabola. The slope of the x vs. t graph equals the instantaneous velocity. The slope of a v vs. t graph equals the acceleration.
  • The slope of the graph v vs. t can be used to find acceleration; the area of the graph v vs. t can be used to find displacement. Welcome to calculus!

What is a Graph

Watch this Explanation

Time for Practice

  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? {\;} d. What is the turtle’s displacement after 4s? {\;}
  2. Draw the position vs. time graph that corresponds to the velocity vs. time graph below. You may assume a starting position x_0 = 0 . Label the y- 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/s^2) 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 , the yellow Maserati starts from rest and accelerates at 10 \ m/s^2 . 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)? {\;} 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)? {\;}

Answers:

1c. 25 m

1d. -5 m

2. discuss in class

3. discuss in class

4b. 3 sec

4d. 6 sec

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