Catapults: Parabolic Motion

Pre-lab

Lead the class in discussing the questions on their worksheet as you go through the demonstration.

Resources Needed:

- Graphing calculator

- Calculator/overhead transparency connection

- Large grid paper (optional)

- Two different color markers

- Soft object for two students to toss to each other (e.g., board eraser, chalk, tennis ball, etc.)

Class Begins

- As students walk into class, you can choose to have the catapults out for them to see or you can pull them out at the end of the pre-lab.

- Have the grid paper already set up on a wall (multiple pages to create a large grid) for a student to draw on. You can use the board if large grid paper is not available.

Creating a Parabola – 3 students

-  Take out the soft object you chose (such as a ball or board eraser) for students to toss to each other.

-  Explain that the class is going to predict path the object is going to take and then they will toss the object to each other and draw the actual path.

1.  A pair of students will toss an object to each other, as directed by your teacher. Before the object is tossed, predict the path of the object and draw it on the grid provided below.

Have two students come and stand in front of the grid paper (or board). Ask a third student to predict the path and draw that prediction on the large grid paper as students draw their predictions in the space provided on their worksheets.

2.  After the object is tossed, draw the actual path in the space provided.

Two students toss object. Make sure students understand to use a slow arcing toss, not like they’re pitching a baseball. The third student can draw the actual path in different color marker as the rest of the class draws the actual path on their worksheet.

PREDICTED PATH ACTUAL PATH

3.  What is this type of path called?

Reinforce that this path is called a parabola.

4.  List at least two real-world examples of parabolas.

Discuss student responses. Some possibilities include basketballs, water from a water fountain, artillery shells, fireworks, and the 360 bridge (in Austin).

5.  Create a table of ordered pairs from selected points on the graph, as directed by your teacher.

Create a data table on the board or graph paper. Pick several points and ask the class to provide the x, y values. Be sure one of the ordered pairs represents the highest point on the parabola.

x / y
1 / 1
4 / 6
9 / 11
15 / 3
16 / 1

6.  Watch as your teacher demonstrates how to use the data to create a graph on the graphing calculator.

Choose 3 points from this table. Some good choices are the x-intercepts and the y-intercept. Be sure one of the points you choose is the highest point on the parabola.

Move your calculator to your overhead. Take students through the steps to move data table into calculator.

You may need to clear previous data out of the calculator first.

A.  Choose the STAT button on you calculator.

B.  In the EDIT menu choose item 1: Edit and press ENTER.

C.  Enter all three points: X coordinates in L1 column, Y coordinates in L2 column.

D.  Choose the 2ND and then the Y= button. That will take you to the STAT PLOT menu.

E.  Make sure that Plot 1 is ON and hit ENTER.

F.  Double check to make sure that Plot 1 is on, that the first plot type is chosen, that the X list says L1 and that the Y list says L2. Choose whichever Mark you would like (these are what the points in the graph will look like).

G.  Press the GRAPH button. The three points should be on the graph. Make sure you can see all three points. If you can’t, choose the WINDOW button and change the range for the x and y axes.

7.  Compare the graph generated by the graphing calculator to the ACTUAL PATH graph you drew. Do they look the same?

The graphs should be similar

8.  Watch as your teacher demonstrates how to generate an equation from the data on the graphing calculator.

Follow these calculator key strokes to create the parabola:

A.  Choose the STAT button again.

B.  In the CALC menu choose item 5: QuadReg and press ENTER L1 (comma) L2 and press ENTER again.

C.  Copy down a, b, and c to make a quadratic. This is your quadratic expression.

D.  Choose the Y= button. Copy the quadratic formula using a, b, and c. (write down the quadratic formula so students see how it is formed from the quadratic regression.

E.  Choose the GRAPH button and a curve should be drawn along the points. Copy that curve onto your graph.

9.  The general from of a quadratic equation is Y = ax2 + bx + c. Write the a, b, and c values for the equation generated by the graphing calculator in the space below.

Y = -0.172X2+2.944X-2.181

Sample equation. Equations will vary.

10. Your teacher will show a toy catapult being launched. How does a catapult work?

Catapults store energy in twisted ropes, a flexed piece of wood, or (as with the toy catapult) a rubber band. When the arm of the catapult is released, it launches the object at the end of rotating arm.

11. What were ancient catapults used for?

Ancient catapults were war machines used to hurl large stones and burning debris. Students may have seen catapults being used in Lord of the Rings: Return of the King or other movies.

For more information on catapults and trebuchets, see http://www.science.howstuffworks.com/question127.htm

12. How could you determine the equation that describes the path of the object launched by the catapult?

Show the catapult and how it is set up for launch. Launch an object so students can see that the path is a parabola. Discuss with students how you can measure the highest point of the parabola and the point where the object lands to get data points. Ask what the starting point of the object would be (0, 0). With three data points, you can use the same calculator procedure to generate the quadratic equation.

13. How could you predict if the object launched from a catapult will hit a given target?

Talk about how students can predict how high or how far an object will fly by using the graph, the table of values, or the quadratic function.

Show an example of solving for y. If the catapult is placed a certain distance from the wall (x) and you want to know how high on the wall the object will hit, you use the equation to solve for y.

14. How will the parabola change if a heavier object is launched?

Get student predictions before launching a heavier object, such as a small piece of chalk or a nickel.

15. How will the parabola change if another rubber band is added on the catapult?

Get student predictions before adding another rubber band (or a stiffer rubber band) and launching an object.

Explain to students that they will be collecting their own data in the lab by “catapulting” objects and taking different measurements. In the post-lab, students will use their data and quadratic equation to hit targets with their catapult.

Catapults: Pre-lab Teacher Notes Page 4 of 1

© 2004 The University of Texas at Austin and the GE Foundation