Today in my Precalculus class, we investigated the connection between projectiles and quadratic functions. We used Pasco launchers to fire steel balls and gather data on their flight paths:

Then we placed ring stands so that the ball passed through them. This gave the students a visual model of the ball’s path:

Next, we discussed the horizontal and vertical components of the ball’s motion, splitting them into separate functions (see my notes below). Then we split up into three groups to calculate the initial velocities of balls launched with one click, two clicks, and three clicks on the launchers.

Students needed to measure the horizontal distance the balls traveled, so they taped carbon paper (which is getting hard to find these days!) to the floor and measured out to the marks made by the balls’ landings.

By averaging the distances from the launcher to the marks, students get a fairly accurate measurement of the horizontal distance traveled.

With this data, they were able to calculate the initial velocity of their projectile!

Here are the notes for the activity:

**Quadratics & Projectiles Lab**

1. Explain that everyone firing the launcher wears safety goggles or glasses!

2. Show the launcher’s markings: height mark, angle of elevation

3. Demonstrate how to load the ball, and difference between clicks

4. Shoot a ball.

*What path did it follow?*

* Where would the vertex of the parabola be?*

5. Students try to set up ring stands to follow the path of the ball. It’s probably helpful to put one ring on a stack of books to spread the rings out more.

*What would make the path look more parabolic?*

6. Use pushing a chair to demonstrate independent forces (push forward while student pushes in a perpendicular direction). *What forces are working independently on the ball?*

7. Set up two launchers, with one cocked at 1-click, and the other cocked at 2-clicks.

*How far will they go?*

* When will they hit the ground?*

(Listen with eyes closed)

*Is motion away from the launcher affected by vertical motion?*

8. Notes:

(a) Horizontal direction (away from launcher):

*What forces are acting in this direction?*

Full version: x = x_{0} + v_{0x}t + ½ a_{x}t^{2} *What is x_{0}? a_{x}?*

x = v_{0}t *What does v _{0} mean?*

(b) Vertical direction (down):

*What forces are acting in this direction?*

*y = y _{0} + v_{0y}t + ½ a_{y}t^{2} What is y_{0} ?(Let it be 0) What is v_{0y}? a_{y}? What is the sign of a? *

*What direction is the ball going at t = 0? (v_{0y} = 0)
*

So, we have two equations:

x = v_{0x}t

y = ½ a_{y}t^{2}

9. Divide into groups: 1-click, 2-click, 3-click. Each group must use a launcher and carbon paper to gather enough data to determine v_{ox}. *Is it necessary to measure time?*

10. Have teams report out results. Question each team about the appropriate use of significant digits considering the way they recorded data (both the height as well as all the horizontal distances).

Special thanks to our Director of STEM education at Harpeth Hall, Dr. Stacy Gardner, for demonstrating this activity to the math department!