The positioning of both feet is essential to good technique in archery and shooting sports, including rifle and pistol. Stance affects balance before, during, and after release of the string or trigger.

Consider the graph below, excerpted from a book series by Schottenbauer Publishing.

__Discussion Questions__
- What is the maximum force during the step?
- What is the mass of the person?
- What is the length of time of the step onto the plate?
- What is the length of time of the step off of the plate?
- Write a linear equation approximating the application of force during the step onto the plate.
- Write a linear equation approximating the application of force during the step off of the plate.
- Describe the effect of stance shifts during archery or shooting sports.

**Additional Information**

Geometry is essential for both archery and rifle sports. Take a moment to write down a few ways in which geometry affects the precision of the sport.

**Discussion Questions**
- What data is necessary to collect in order to understand the role of geometry in shooting?
- What spatial perspectives and/or mathematical planes are important for precision?

The cover of *The Geometry of Summer Olympic Sports*, to the right above, features an archer taking aim.

**Discussion Questions**
- What angles can be measured on the diagram, in order to understand the accuracy of technique? The path of the arrow?
- Is any essential information missing from the picture? What is necessary in order to measure that information?

Geometry diagrams featuring archery and shooting sports are available in the following books from **Schottenbauer Publishing**:

**Geometry Workbooks**

**Additional Information**
The position of a rifle or bow is essential to sport shooting, whether with stationary or moving targets. Graphing the position and movement of a rifle or bow provides an interesting activity for classes in physical education, math, and science.

Consider the graph below, excerpted from *The Science of Archery & Shooting Sports: Volume 2* from Schottenbauer Publishing.

Discussion Questions
- Sketch the position of the gun at the following points: (a) Initial Position, (b) Any Intersecting Lines, (c) Final Position.
- How far does the nozzle travel in the x direction? In the y direction?
- How far does the butt travel in the x direction? In the y direction?
- Describe the shape of motion.
- What is the average speed of the nozzle in the x direction? in the y direction?
- What is the average speed of the butt in the x direction? in the y direction?
- Is this most likely a toy rifle or a toy handgun?

Graphs and data on archery and shooting sports can be found in the following science lab manuals from Schottenbauer Publishing:

Graphs & Data for Science Lab: Multi-Volume Series
Anthologies of 28 Graphs

**Additional Information**
In archery and shooting sports, a tiny difference of aim can result in a big difference at the target. Consider the trajectories of two arrows below, both excerpted from *The Science of Archery & Shooting Sports: Volume 1* from Schottenbauer Publishing.

Discussion Questions

- Which graph shows an arrow moving in a straight line?
- Which graph shows an arrow moving up and down?
- Why are points missing in the second graph? Sketch in the trajectory which is most likely during this segment of time.
- Redraw each graph using a similar scale on each axis.
- Redraw each graph using a similar scale on each axis, and re-labeling the origin of the tip of the arrow at x=0, y=0, and t=0.
- Draw the motion of each arrow on a separate piece of paper, using the information from #5 above.
- On the drawings from #6 above, add labels showing time at (a) release from the bow, (b) apex, (c) final point.
- Describe the range of x, y, and t for each graph, using the information from #5.
- Which arrow travels further in the x direction? In the y direction?
- Describe the absolute distance which each arrow travels. Which arrow travels further, in absolute distance?
- Which arrow is in flight for a longer period of time?
- For each graph, draw the arrow in starting position. Using a compass, measure the angle.
- Which arrow was aimed better? What information is necessary to answer this question?

Graphs and data on archery and shooting sports can be found in the following science lab manuals from Schottenbauer Publishing:

Graphs & Data for Science Lab: Multi-Volume Series
Anthologies of 28 Graphs
**Additional Information**
What is the balance point of an arrow, and why does it matter? The data below (Copyright 2014, All Rights Reserved), excerpted from the book series *The Science of Archery & Shooting Sports: Volume 1* from Schottenbauer Publishing, describe the balance point of a toy arrow.

Discussion Questions
- How does the arrow sit on the bow? Draw a sketch of the arrow in the bow, in the following positions:
- Bowstring loose
- Bowstring drawn to maximum

- How large is the arrow, proportional to the bow? Describe the following proportions. Are there any other proportions which might be relevant to understanding the bow and arrow?
- Arrow length divided by bow width
- Arrow length divided by bow height
- Arrow mass divided by bow mass

- What is the balance point of the arrow? How does the balance point affect the trajectory of the arrow?
- What would occur if the arrow were weighted towards either end?

Graphs and data on archery can be found in the following science lab manuals from Schottenbauer Publishing:

Graphs & Data for Science Lab: Multi-Volume Series
Anthologies of 28 Graphs

**Additional Information**
The tension of a bow string provides the force which propels arrows long distances. The bow string acts as a spring, with properties which can be analyzed in a method similar to classical physics experiments.
The graph below (Copyright 2014, All Rights Reserved), excerpted from the book series *Gravity, Springs, & Collisions: Graphs from Classical Physics Experiments* from Schottenbauer Publishing, shows an experiment which has physical properties very similar to a pull on a bow string.

Discussion Questions
- What is the maximum force on the spring during each of the 3 pulls?
- In what direction(s) do the wireless device accelerate?
- If a bow string were to possess the same tension as this spring, what would be the resulting acceleration of an arrow with 0.03 kg mass? Calculate the acceleration for each of the maximum forces above.
- How is this experiment similar to the pull on a bow string? How is it different?

Additional graphs similar to those above can be found in the following science lab manuals from Schottenbauer Publishing:

Graphs & Data for Science Lab: Multi-Volume Series
Anthologies of 28 Graphs
**Additional Information**