Lesson 1 - 2D Uniform Motion Builder Graphing (pos, vel)

2D Uniform Motion Builder Graphing (pos, vel) simulates the uniform motion of an object and constructs position-time and velocity-time graphs to analyze one-dimensional motion.

Prerequisites

Students should have completed Lesson 1 Motion Builder 1D - Position and Velocity Graphs. They will require a working knowledge of the vector quantities position, displacement, and velocity. Students should also be familiar with basic graphing skills including slope and area calculations.

Learning Outcomes

Students will be able to use graphical analysis to describe one-dimensional motion. Specifically, they will be able to generate and interpret position-time, velocity-time and acceleration-time graphs. Students will also be able to use slope and area calculations to generate data and solve problems.

Instructions

Students should understand the applet functions that are described in Help and ShowMe. The applet should be open. The step-by-step instructions on this page are to be done in the applet. You may need to toggle back and forth between instructions and applet if your screen space is limited.

Contents

  1. Position, Velocity, and Acceleration: Slope Relationships
  2. Position, Velocity, and Acceleration: Area Relationships
  3. Summary of Slope and Area Relationships
  4. Applications to Complex Motion Problems

1. Position, Velocity, and Acceleration: Slope Relationships

shuttle.gif

The space shuttle launch is an example of non-uniform (accelerated) motion. Graphical analysis of the shuttle's position with respect to time will reveal the relationships between position, velocity and acceleration.

exercise 1

Use the 2D Uniform Motion Builder Graphing (pos, vel) applet to generate a position-time graph for the shuttle launch, which is initially at rest (v = 0.0 m/s) and accelerates at 5.0 m/s2 for 10.0 s.

  1. On the applet, click "Reset" (Reset) and then "Add" (Add) to add an object.

  2. In the "EditorDialog" window enter a time of 10.0 s, an initial velocity of 0.0 m/s and an acceleration of 5.0 m/s2.

    motion_step_1.gif (3939 bytes)

  3. Click "Play" (Play) and observe the moving object (object.gif (1651 bytes)) accelerate.

  4. To view the position-time graph:

  1. To view the slope, click "Find Slope" (slope.gif (1719 bytes)). Click and hold the mouse over the graph line and the slope value (m/s) will be indicated under "output".

Verify that your graph is identical to Figure 1. The slope value is displayed under "output" and represents the shuttle's instantaneous velocity at the selected time. For example, Figure 1 shows an instantaneous velocity of +25.0 m/s at 5.0 s.

Figure 1: Position-Time

figure_1.gif (8107 bytes)

exercise 2

Explain how you can tell, by visual inspection, that the motion of the shuttle is not uniform, but is speeding up. In your explanation refer to the term slope.
exercise 3

Use the position-time graph created in Exercise 1 and the "Slope" tool (slope.gif (1719 bytes)) to complete the following tasks.

  1. Complete the following table. (Double-click on "Slope" (slope.gif (1719 bytes)) to enter the exact time value for the point you wish to examine. The velocity will be displayed under "output".)
Time (s) Velocity (slope) m/s
0 _______
2 _______
4 _______
6 _______
8 _______
  1. Using the velocity data collect in part (a), complete the graph below.

Velocity vs. Time

graph_paper.gif (5128 bytes)
exercise 4

Which of the following best describes Graph 1: Velocity-Time? Write an equation expressing the relation between velocity and time.

  1. The graph is constant and of the mathematical form image, b is a constant.
  2. The graph is linear and of the mathematical form image, b is a constant and m is the slope.
  3. The graph is a quadratic curve and of the form image, where a, b, and c are coefficients.
It is clear from the previous exercises that the velocity of the shuttle discussed above is not constant. The velocity is continuously changing from a value of 0.0 m/s to a value of 40.0 m/s after 8.0 seconds. There is clear evidence that the shuttle is accelerating.

exercise 5

Use the applet to create a velocity-time graph based on the same time, velocity, and acceleration values that were entered in Exercise 1 by doing the following:

  • Change the "y-axis" to show "velocity" (y_vxo.gif (2013 bytes))
  • click "Fit Graph" (fit.gif (1665 bytes))
  1. Verify that this graph is identical to Graph 1: Velocity-Time that you created in Exercise 3.

  2. Calculate the slope and verify your answer using the slope tool on the applet.


  3. What are the units that correctly describe the slope of this graph?


  4. What does the slope "mean" here? That is, what quantity of motion does the slope measure?
exercise 6

Use the applet to create an acceleration-time graph based on the same time, velocity, and acceleration values that were entered in Exercise 1 by doing the following.

  • Change the "y-axis" to show "acceleration" (y_ax0.gif (2033 bytes))
  • click "Fit Graph" (fit.gif (1665 bytes))
  1. Complete the graph of Acceleration vs. Time.

Acceleration vs. Time

graph_paper.gif (5128 bytes)

  1. In Exercise 5, you determined the slope of the velocity-time graph for the shuttle launch. Where or how does that "appear" on the acceleration-time graph?



  2. Is the slope of Graph 2 equal to zero? What does this mean?


2. Position, Velocity, and Acceleration: Area Relationships

In this section, we will explore the idea that area is a measure of the total or cumulative change in some variable. The variables will include position, velocity, and acceleration.

exercise 7

Fill in the blanks using the terms position, velocity, or acceleration.

  1. A change in position in an interval of time is called ______________.

  2. A velocity acting for an interval of time produces a change in ______________.

  3. A change in velocity in an interval of time is called ______________.

  4. An acceleration acting for an interval of time produces a change in _____________.

exercise 8

Use the applet to generate an acceleration-time graph for a ball falling from rest (v = 0.0 m/s) for 5.0 seconds.

  1. Clear all objects from the applet by selecting them and clicking "Remove" (minus.gif (1658 bytes)).

  2. Click "Reset" (Reset) and then "Add" (plus.gif (1723 bytes)) to add an object.

  3. In the "EditorDialog" window enter a time of 5.0 s, an initial velocity of 0.0 m/s and an acceleration of -9.81 m/s2.

  4. Click "Play" (Play) and observe the moving object (object.gif (1651 bytes)) accelerate.

  5. Generate an acceleration-time graph and click "Fit Graph" (fit.gif (1665 bytes)).

Verify that your graph is identical to Figure 2: Acceleration-Time.

Figure 2: Acceleration-Time

image

exercise 9

Using the "Area" tool (image), sweep out the area (from left to right) between t = 0.0 s and each of the following times. (Double-click on "Area" (image) to enter the exact time value for the point you wish to examine. The area should be green and will be displayed under "output".)

Time (s)

Area Image

Area (Output)

1.0

ex9_interval_1.gif (3148 bytes)

_______

2.0

ex9_interval_2.gif (3085 bytes)

_______

3.0

ex9_interval_3.gif (3115 bytes)

_______

4.0

ex9_interval_4.gif (3194 bytes)

_______

5.0

ex9_interval_5.gif (2903 bytes)

_______
exercise 10

Each of the areas that were measured are rectangles. The area of a rectangle is just "length x height".

  1. What are the units of "length" for each of these rectangles?

  2. What are the units of "height" for each of these rectangles?

  3. Multiply the units of length and height to determine the units of "area" for each of the rectangles. What variable is the area describing?
The results of Exercise 10 show that the area of an acceleration-time graph indicates the total change in the velocity of the motion. The same is true when relating velocity and position.

exercise 11

Use the applet to create a velocity-time graph based on the same time, velocity, and acceleration values that were entered in Exercise 8 by doing the following.

  • Change the "y-axis" to show "velocity" (y_vxo.gif (2013 bytes))
  • click "Fit Graph" (fit.gif (1665 bytes))

  • Verify that your graph is identical to Figure 3: Velocity-Time.

Figure 3: Velocity-Time

image
exercise 12

Using the "Area" tool (image), sweep out the area (from left to right) between t = 0.0 s and each of the following times. (Double-click on "Area" (image) to enter the exact time value for the point you wish to examine. The area should be green and will be displayed under "output".)

Time (s)

Area Image

Area (Output)

1.0

ex12_interval_1.gif (3096 bytes)

_______

2.0

ex12_interval_2.gif (3194 bytes)

_______

3.0

ex12_interval_3.gif (3291 bytes)

_______

4.0

ex12_interval_4.gif (3350 bytes)

_______

exercise 13

Each of the areas that were measured are triangles. The area of a triangle is just 1/2 "length x height".

  1. What are the units of "length" for each of these triangles?

  2. What are the units of "height" for each of these triangles?

  3. Multiply the units of length and height to determine the units of "area" for each of the triangles. What variable is the area describing?
exercise 14

Plot the data collected in Exercise 12. Explain what this graph shows. Why are all of the positions "negative"?

Position vs. Time

graph_ex_14.gif (5936 bytes)

3. Summary of Slope and Area Relationships

The slope and area properties of graphs are related to the actual properties of motion. This is summarized in the table below.

Slope Properties

  1. The slope of a position-time graph is a measure of instantaneous velocity.
  2. The slope of a velocity-time graph is a measure of instantaneous acceleration.

Figure 4: Slope Summary

slope_summary.gif (9810 bytes)

Area Properties

  1. The area of an acceleration-time graph is a measure of the total change in velocity.
  2. The area of a velocity-time graph is a measure of the total change in position.

Figure 5: Area Summary

area_summary.gif (11202 bytes)

4. Applications to Complex Motion Problems

Consider some complex motion situations in which an object undergoes several different accelerations. Use 2D Uniform Motion Builder Graphing (pos, vel) to assist you in answering these questions.

exercise 15

You are riding in an elevator. Starting from rest, the elevator undergoes the following motions.

  • It accelerates from rest (v = 0.0 m/s) upwards for 5.0 s at +2.0 m/s2.

  • It then coasts for 20.0 s at 10.0 m/s (a = 0.0  m/s2).

  • Finally, starting at 10.0 m/s, it accelerates downward for 2.5 s at -4.00 m/s2.

Complete the following graphs:

Position vs. Time

graph_paper.gif

Velocity vs. Time

graph_paper.gif

Acceleration vs. Time

graph_paper.gif

  1. Using the slope of the position-time graph, determine the maximum speed reached by the elevator.




  2. Using area, determine how far the elevator traveled while coasting.





  3. Using area, determine how far the elevator traveled while accelerating downward.
exercise 16

Brenda is standing on the edge of a cliff and tosses her physics book upward with a speed of 22.0 m/s. It hits the ground at the base of the cliff 6.0 s later. Use 2D Uniform Motion Builder Graphing (pos, vel) to determine how high the cliff is and how fast the book was moving when it landed.

exercise 17

You are programming a control computer for a subway train with the following parameters.

Use the following questions and the applet to help create motion scripts that could accomplish this.

  1. How much time is required to accelerate from rest at 2.00 m/s2 in order to achieve the maximum speed?



  2. How far does the subway train travel while accelerating up to the maximum speed?



  3. How long will it take the train to stop if it starts at the maximum speed and accelerates at 3.00 m/s2?



  4. How far does the subway train travel while stopping?



  5. Given the starting and stopping distances (b and c), how far and how long does the train travel at the maximum speed?



  6. How much time will it take for the subway train to travel between the stops?



  7. What is the average speed of the train as it travels between stops?




  8. Give the three motion scripts which will be used to control the train:

    1. time: ______ s , initial velocity ______ m/s, acceleration: ______ m/s2

    2. time: ______ s , initial velocity ______ m/s, acceleration: ______ m/s2

    3. time: ______ s , initial velocity ______ m/s, acceleration: ______ m/s2

  9. Draw the following graphs representing the train's motion.

Position vs. Time

graph_paper.gif (5128 bytes)

Velocity vs. Time

graph_paper.gif (5128 bytes)

Acceleration vs. Time

graph_paper.gif (5128 bytes)

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Last Updated: June 16, 2004