Trajectories of human motion can be analyzed in several ways. Most often, scientists are interested in measuring the simple velocity or speed of a human walking, jogging, or running forward on a track. A closer analysis of human motion shows a complex pattern of movements, however. As the torso moves forward, the legs and arms move back and forth relative to the torso. In addition, the entire body bounces slightly up and down. The force of breathing can also distort the shape of the torso.

What does human movement look like in a graph? Consider the graph below, excerpted from the book series *Walk, Jog, & Run: The Science of Athletic Training* from Schottenbauer Publishing.

__Discussion Questions__
- In which direction is the walker moving?
- How many steps are shown in the graph?
- How can the speed of the walker be measured on the graph?
- Describe the movement of the knee, ankle, and toes relative to the hip, as if the hip were stationary.
- Which body part moves the most in the x direction?
- Which body part moves the most in the y direction?
- What is the velocity of the walker?

**Additional Information**
Geometry is essential for physical fitness. Take a moment to write down a few ways in which geometry affects the precision of sport.

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

The cover of *The Geometry of Water Sports*, to the right above, features a waterskier in action.

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

Geometry diagrams featuring physical fitness activities are available in the following books from Schottenbauer Publishing, available in paperback and Kindle editions:

**Geometry Workbooks**

**Additional Information**
Raw data sometimes consists of individual points of information which are difficult or impossible to connect in a meaningful way. Scientific processes require collecting data in a purposeful manner, to elucidate the connections between different phenomena.

Consider the following graph, excerpted from the series The Science of Exercise Equipment from Schottenbauer Publishing:

__Discussion Questions__
- Connect the dots in the graph, to reveal the pattern of motion of the bike pedal.
- Draw the bike pedal in motion, labeling at least 4 points in time.
- What are the minimum and maximum x-positions of the pedal?
- What are the minimum and maximum y-positions of the pedal?
- How many pedal rotations are shown in the graph?
- What is the average time of one rotation?
- What occurs at the end of the graph?

The following books from Schottenbauer Publishing contain a more extensive collection of graphs and data pertaining to the science of athletic training:

*Graphs & Data for Science Lab: Multi-Volume Series*
- The Science of the Athletic Training
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Exercise Equipment
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Gymnastics
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Yoga, Pilates, & Ballet
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

Bouncing up and down is a healthy activity that stimulates the lymphatic system. The shape of bouncing is determined in part by the act of pushing up from the ground, as well as the effect of the force of gravity on the body, and any sort of cushioning of the descent by the joints or equipment.

Consider the following graph, excerpted from the series The Science of Exercise Equipment from Schottenbauer Publishing:

__Discussion Questions__
- How high is the bounce?
- Over what time segment does the bounce occur?
- Does the person travel sideways during the bounce?
- Is it possible to determine whether the bounce is cushioned by the knees, the Bosu, or both? If so, what provides the most cushioning? [Note: A Bosu is a dome-shaped inflatable device used for exercise.]
- On a piece of paper, sketch the person's position at the beginning, highest point, and end of the bounce. Focus on the center of the abdomen.
- Redraw the graph with the initial position at (0,0). Label the axes with the correct coordinates.
- Redraw the graph with the highest bounce position at (0,0). Label the axes with the correct coordinates.
- Redraw the graph with the horizontal line on the x-axis and the highest bounce position at x=0. Label the axes with the correct coordinates.
- On a piece of paper, list as many athletic activities as possible which involve bouncing or jumping. Use a time limit of 60 seconds.

The following books from Schottenbauer Publishing contain a more extensive collection of graphs and data pertaining to the science of athletic training:

*Graphs & Data for Science Lab: Multi-Volume Series*
- The Science of the Athletic Training
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Exercise Equipment
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Gymnastics
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Yoga, Pilates, & Ballet
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

**Additional Information**
Repeating patterns are common both in math and in exercise science. Consider the following graphs, excerpted from the series The Science of Exercise Equipment from Schottenbauer Publishing:

__Discussion Questions__
- What type of motion is described the graph?
- What are the maximum and minimum x positions of the pedal? The y positions? Write these as coordinate pairs (x, y).
- Sketch the real motion of the pedal, indicating at least 8 equally-spaced points in time.
- What is the period (length) of the cycle?

__Discussion Questions__
- What type of motion is described the graph?
- What are the maximum and minimum angles of the arm?
- Sketch the real motion of the arm, indicating at least 8 equally-spaced points in time.
- What is the period (length) of one cycle?

__Discussion Questions__
- What phenomenon is described in the graph?
- What are the minimum and maximum values of potential?
- Is a cycle described in the graph? If so, describe the cycle in words, relating potential to exercise activity.
- What is the period (length) of one cycle?

__Discussion Questions__
- What phenomenon is described in the graph?
- What are the minimum and maximum values of flow rate?
- What is the average minimum value of flow rate? The average maximum value?
- Is a cycle described in the graph? If so, describe the cycle in words, relating flow rate to behavior.
- What is the period (length) of one cycle?

The following books from Schottenbauer Publishing contain a more extensive collection of graphs and data pertaining to the science of athletic training:

*Graphs & Data for Science Lab: Multi-Volume Series*
- The Science of the Athletic Training
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Exercise Equipment
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Gymnastics
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Yoga, Pilates, & Ballet
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

**Additional Information**
How much sport science can be crammed into only 28 graphs? This question inspired a series of anthologies by Schottenbauer Publishing, spanning 10 popular Olympic and common sport topics.
__Sport Science Anthologies__
- Physical Fitness
- Summer Olympic Sports
- Winter Olympic Sports
- Gymnastics
- Track & Field
- Ball Sports
- Dance & Ballet
- Yoga
- Figure Skating
- Ice Hockey

__Video Analysis__

- Jumping Jack
- Pedaling on Bike

__Goniometer__
- Seated Knee Extensions
- Jogging in Place

__Spirometer__

- Normal Breathing
- After 5 Minutes Jogging (& Continuing to Jog)

__Electromyogram__

- Arm Relaxed
- Arm Held Straight Up
- Arm while Jogging
- Arm during Bicep Curls
- Quadricep During Vertical Squat

__Electrocardiogram__

- Before Cycling
- After 5 Minutes Cycling
- 4 Minutes After Ending Cycling

__Discussion Questions__

- What comparisons and contrasts can be made with this data?
- What additional data would be useful to obtain?

The following books from Schottenbauer Publishing contain a more extensive collection of graphs and data pertaining to the science of athletic training:

*Graphs & Data for Science Lab: Multi-Volume Series*
- The Science of the Athletic Training
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Exercise Equipment
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Gymnastics
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

- The Science of Yoga, Pilates, & Ballet
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

**Additional Information**
Bicycling provides excellent examples of motion for physics education. Road bikes demonstrate a combination of translational and rotational motion, while stationary bicycles demonstrate the rotational motion of the wheel.

The graph below, excerpted from *The Science of Exercise Equipment, Volume 1*, shows the acceleration, force, and altitude changes associated with the rotational motion of a recumbent stationary bicycle.

Discussion Questions
- What are the minimum and maximum points on each graph?
- Why does the graph show acceleration in three directions of motion? Describe how each component, including x, y, and z, relates to the motion of the ankle.
- What aspect of bicycle motion is normally modeled by a sine wave?
- Can any of these graphs be modeled with a sine wave? Why or why not?
- What is the period and frequency of the motion in the graph?
- Does the athlete change speed during the motion recorded in the graph? Why or why not?
- What is more useful for analyzing the motion of the wheel: a) the total force, b) the x, y, and z acceleration? Why?
- Is the athlete pedaling forwards or backwards?

The free YouTube video, *Understanding the Motion of the Wheel* from **Schottenbauer Publishing, **provides graphical analysis of video footage. In the video, spatial analysis of motion (e.g., what does the motion look like to a viewer) is compared to graphical analysis of motion (e.g., what does the motion look like in a graph). This video can supplement traditional lectures on the science of basic motion. The video, shown below, is discussed in detail in the blog post Understanding Translational and Rotational Motion from a Bicycle Wheel on the blog The Science of Transportation.

The following books from Schottenbauer Publishing contain similar types of graphs and data pertaining to the science of bicycles:

*Graphs & Data for Science Lab: Multi-Volume Series*
- The Science of the Wheel
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Video Analysis, Force, Velocity, Acceleration
- Volume 3: Video Analysis, Force

- The Science of Exercise Equipment
- Volume 1: Force, Velocity, Acceleration
- Volume 2: Biophysics
- Volume 3: Video Analysis
- Volume 4: Video Analysis

**Anthologies of 28 Graphs**
- The Science of Transportation
- The Science of Physical Fitness
- The Science of Summer Olympic Sports

**Additional Information**