1

Biomechanics of Free-Throw Motion—Uniform Motion and Lever

Mechanics

Brian Fox

Student at Lewis and Clark State College

Lewiston, Idaho

Erik Helleson

Cheney High School

Cheney, WA

Bill Knapp

Timberline High School

Weippe, Idaho

Eric Nordquist

Colton Middle School

Colton, WA

Glenn Voshell

Colton High School

Colton, WA

Washington State University Mentor

Professor David Lin

Bio-Engineering

July, 2008

The project herein was supported by the National Science Foundation Grant No. EEC-

0808716: Dr. Richard L. Zollars, Principal Investigator and Dr. Donald C. Orlich, co-PI.

The module was developed by the authors and does not necessarily represent an official

endorsement by the National Science Foundation.

2

Table of Contents

Page #

Summary 3

Introduction 3

Rationale 5

Science 5

Engineering 5

Goals 6

Prerequisite Skills / Knowledge 6

Equipment (model construction) 8

Activities 11

Lever Mechanics of the Throwing Motion

Lever Mechanics Student Worksheet

Finding the Velocity of a Projectile Using Vectors

Vector Activity Student Worksheet

Uniform Motion Activity

Uniform Motion Activity Student Worksheet

Extensions 26

Appendices 27

The Quadratic Formula

The Pythagorean Theorem

Right Triangle Trigonometry

Uniform Motion Problems

Vector Algebra

References 34

3

Summary

Overview of Project

This module has been designed to introduce Jr. High School and Senior High School students to

the field of engineering and to enhance their interest in engineering and its practical application

through bioengineering. The module will first look at the anatomy/physiology of the human arm

and more specifically, the elbow and wrist joints by using a simple, homemade model. Then,

using the arm model as a springboard, the module will explore the anatomical and physiological

aspects of the arm including: 1), the basic skeleton-muscular structures of the arm; 2) the lever

mechanics of how muscles work together to perform an action of shooting a basketball free

throw using a projectile; 3) the algebra and trigonometry involved in how the arm moves at

specific angles and how changing insertion points of various muscles might impact the flight of

the ball

Intended Audience

As mentioned earlier, our intended audience will be middle school and high school students. The

basic skeleton-muscular structures of the arm including muscles, bones, tendons, ligaments can

be taught in middle school life-science. The algebra segments of this module can be incorporated

in 9th

grade physical science with the building of the model and the performing of the throwing

function. More detailed anatomy can be taught in high school anatomy classes and high school

trigonometry classes can investigate angles, velocities and vectors.

Estimated Duration

The duration of these activities could run from two or three days to three weeks, depending on

the activities chosen.

Introduction

Biological engineering is any type of engineering--for example, mechanical engineering--applied

to living things.

Bioengineers are concerned with the application of engineering sciences, methods, and

techniques to problems in medicine and biology. Bioengineering encompasses two closely

related fields of interest: the application of engineering sciences to understand how animals and

plants function; and the application of engineering technologies to design and develop new

devices, including diagnostic or therapeutic instrumentation, or the formulation of synthetic

biomaterials, the design of artificial tissues and organs, and the development of new drug

delivery systems.

Bioengineering is the application of the principles of engineering and natural sciences to tissues,

cells and molecules. Closely related to this is biotechnology, which deals with the

implementation of biological knowledge in industrial processes. Applications from both fields

are widely used in medical and natural sciences and also in engineering.

4

The National Institute of Health (NIH) Bioengineering Consortium agreed on the following

definition for bioengineering research on biology, medicine, behavior, or health recognizing that

no definition could completely eliminate overlap with other research disciplines or preclude

variations in interpretation by different individuals and organizations.

Definition

Bioengineering integrates physical, chemical, or mathematical sciences and engineering

principles for the study of biology, medicine, behavior, or health. It advances fundamental

concepts, creates knowledge for the molecular to the organ systems levels, and develops

innovative biologics, materials, processes, implants, devices, and informatics approaches for the

prevention, diagnosis, and treatment of disease, for patient rehabilitation, and for improving

health.

If we ignore the obvious health focus in the NIH definition, it is clear that bioengineering is

concerned with applying an engineering approach (systematic, quantitative, and integrative) and

an engineering focus (the solutions of problems) to biological problems.

“Bioengineering or Biomedical Engineering is a discipline that advances knowledge in

engineering, biology, and medicine --and improves human health through cross-disciplinary

activities that integrate the engineering sciences with the biomedical sciences and clinical

practice. Major advances in Bioengineering include the development of artificial joints, magnetic

resonance imaging (MRI), the heart pacemaker, arthroscopy, angioplasty, bioengineered skin,

kidney dialysis, and the heart-lung machine (“Bionewsonline,” 2005).

Preparation

A bachelor‟s degree in engineering is required for almost all entry-level engineering jobs.

Unlike many other engineering specialties, a graduate degree may be recommended or

required for some entry-level jobs in bioengineering. College graduates with a degree in a

physical science or mathematics occasionally may qualify for some engineering jobs,

especially in specialties in high demand. Most engineering degrees are granted in electrical,

electronics, mechanical, chemical, civil, or materials engineering. However, engineers trained

in one branch may work in related branches. For example, some biological engineers also

have training in mechanical engineering. This flexibility allows employers to meet staffing

needs in new technologies and specialties in which engineers may be in short supply. It also

allows engineers to shift to fields with better employment prospects or to those that more

closely match their interests (Sloan Career Center, 2003).

5

Rationale

One of the goals of the Washington State University and National Science Foundation Institute

for Science and Mathematics Education through Engineering Experiences is to have participants

prepare a teaching module that is appropriate for their classroom. These activities will help to

illustrate the connection between science and engineering. Students will be shown engineering

principles that are applied in a diagnostic laboratory. This module is based on research being

done in the Bio-Engineering Department at Washington State University under the direction of

Professor David Lin. The focus of his work is muscle physiology and how muscles can act as

springs or dampers, depending on the situation.

Science

Science can be defined as “accumulated and established knowledge, which has been

systematized and formulated with reference to the discovery of general truths or the operation of

general laws; knowledge classified and made available in work, life, or the search for truth;

comprehensive, profound, or philosophical knowledge.” Science is generally driven by the quest

to find out why something happens. A method of learning about the natural world, science

focuses on formulating and testing naturalistic explanations for natural occurrences.

Engineering

“Engineering is the application of science to the needs of humanity. This is accomplished

through knowledge, mathematics, and practical experience applied to the design of useful objects

or processes. Professional practitioners of engineering are called engineers. Engineering is

concerned with the design of a solution to a practical problem. A scientist may ask „why?‟ and

proceed to research the answer to the question. By contrast, engineers want to know how to solve

a problem, and how to implement that solution. In other words, scientists investigate phenomena,

whereas engineers create solutions to problems or improve upon existing solutions. However, in

the course of their work, scientists may have to complete engineering tasks (such as: designing

experimental apparatus, or building prototypes), while engineers often have to do research. In

general, it can be stated that a scientist builds in order to learn, but an engineer learns in order to

build.” [6]

6

Module Goals and Objectives

The students will build a model of a human arm under direction from the teacher

The students will analyze the dynamics of the human arm and its musculature

The students will use biological methods to understand the muscle structure in the human

arm

The students will see and understand the relationship between engineering and the

biological sciences

Skills / Background Knowledge

This module is not designed to be a standalone exercise. It is strongly recommended that the

students have a background in the following mathematical and physical concepts:

Right Triangle Trigonometry

Quadratic Equations

Vector Algebra

Levers

7

Frayer Model

When a teacher elects to use this module, it is helpful to know the students background

knowledge. This can be achieved using the Frayer Model (Barton, ML & Heidema, C, 2002).

The time required to administer this assessment is approximately 10 minutes. The Frayer Model

has the following characteristics: It is a diagram that is divided into four parts. The teacher

inserts the word / concept to be assessed in the center circle. The student then fills in the

remaining quadrants as illustrated in the diagram provided. Once the students are finished , the

teacher can then assess the students level of understanding in a very short time and cover any

concepts students may have difficulty with.

8

Free Throw Model Construction

Materials:

¼” Dowel

3/16” dowel

1/4” Threadstock- 6.5” length

6 ¼” nuts

2 ¼” washers

1 3” bolt

1 2” bolt

2X6 38cm length

4 eye hooks

25 cm oxyacetylene welding rod or stiff wire (ie metal coat hanger)

2 5cm diameter wooden pulley wheels

2 2X.5X20cm wood scrap (paint stirring sticks)

80 cm of 2X2 cm wood scrap (maybe garden posts)

Rubber bands (variety)

8 feet of string

3” wood screw

Any epoxy or adhesive, ie JB Quik Weld

Practice golf ball

Tools (Jigsaw, plyers, table saw)

Dimensions Wrist: 2X2cmX5cm scrap wood

1cm top and 1 cm bottom eye-hook for tendon insertion

12 cm welding rod for hands (2)—any stiff wire (like a coat hanger)

wood screw centered on wrist block

Adjustable Stop: 1X1, at least 5 cm long (Paint stick or pencil will work)

Radius and Ulna: 20 cm length each

Elbow hinge bolt: 2 cm from end

Wrist hinge bolt: 1.5 cm from end

Insertion points: 7 cm, 10 cm, 13 cm

Humerus: 30 cm length

Shoulder hinge bolt: 2 cm from end

Elbow hinge bolt: 2.5 cm from end

Muscle insertion holes: 6 and 9.5 cm from shoulder end

Muscle insertion to wrist joint connection: 7 cm from elbow end (top & bottom)

Base: 38 cm length

Cut 6x12cm center end

Holes on each side for dowels: 5, 8, 11 cm

9

Construction:

1. Base: Cut 6X12cm slot at center on one end

2. Drill 5/16” hole 3cm from cut-end centered into both sides

3. Drill ¼” stop holes at 5, 8, 11 cm centered on both sides of slot

4. (optional) With 1” drill bit, approx 1/8” deep for golf ball holder, 5cm centered from end

5. Humerus: Cut 30cm length of 2cmX2cm wood

6. Shoulder hinge: Drill ¼” 2 cm from one end on side for threadstock

7. Muscle insertion holes: Drill ¼” 6cm & 9.5cm from shoulder end

8. Elbow hinge: Drill ¼” on side 2.5 cm from end.

9. Cross-joint muscle eye hooks: Eye hooks top & bottom 7cm from elbow end

10. Radius & Ulna: Cut 20cm lengths each of 2X.5 scrap wood (paint stirrers)

11. Drill ¼” 2cm from elbow end

12. Drill ¼” 1.5cm from wrist end

13. Drill 7/32” at 7cm, 10cm, & 13cm for insertion points

14. Cut 3/16” dowel 6 cm in length

15. Wrist: Cut 5cm length of 2X2

16. Drill ¼” 1.5 cm from wrist end

17. Drill holes ~ 1cm from fingers end for the fingers (do not drill all the way through)

18. Screw 3” wood screw into fingers end (adjustable)

19. Screw in eye hooks centered over wrist joint (top & bottom) for tendon insertions

20. Assembly: Insert threadstock through base

block and the humerus, making sure to put

nuts and washers on inside flanking humerus.

Tighten to center humerus in slot (make sure

washers are against wood) See picture to left.

21. Screw on ¼” nuts to hold threadstock in place

on base

22. Insert ¼” dowel into humerus

23. Insert wood pulleys in between humerus and

radius/ulna. See picture below.

24. Insert 3” bolt and fasten lightly

25. Put wrist block in between radius and

ulna and insert 2” bolt & fasten lightly

(make sure it is not torqued too much)

See Picture below.

10

26. Insert oxyacetylene welding rods into arc- into

pre-drilled holes and bend accordingly with

plyers. Epoxy. For safety put on optional

nalgene tubing into end of rods (Thank you

Mr. N). See picture on right.

27. Insert 3/16” dowel into pre-drilled holes of

radius/ulna

28. Cut string into 2 ~18 cm lengths. At one end

tie to larger rubber bands. To other end tie to

3/16” dowel

29. Cut string (make sure you have enough for ease of tying the knots). Tie one end to top

eye hook of humerus and at the other end to the bottom eye hook on wrist block. Assure

that when flexed the tendon string stops the arm from folding. See picture bottom left.

30. Run rubber band through top eye hook of humerus and

loop through itself. Tie piece of string to r.b. and attach

to top eye hook of wrist block. Make sure to put tension

on r.b. and tie string off.

31. Same instructions as 30, except rubber band attaches to

bottom eye hook of humerus and the string to bottom eye

hook of wrist.. See picture below right.

32. Insert ¼”

dowels into

base and

insert stop

block.

33. Enjoy.

Safety: This is a tension-filled model that carries the possibility of rubber bands snapping, thus

it would be strongly advised that all operators wear proper eye protection.

11

Lever Mechanics of the Free-Throw Motion

The lever is often the first example of a simple machine that is taught in an introductory science

class. A lever is defined as a rigid bar that is free to move around a

fixed point. Each lever is composed of different parts: The bar

rotates around a fulcrum, the input force is the force done on the

machine, and the output force is the force exerted by the machine.

There are three different classes of levers, each differentiated by the

different locations of the fulcrum, input and output forces as shown

in the figure to the left.

Activity Objective: After completion of this activity the students

will be able to identify the types of levers present in the free-throw

motion, noting the location of the fulcrum, input and output forces,

and deduct which muscle insertion provides the greatest mechanical

advantage. Finally, the students will investigate the scientific

method fallacies of this exercise and re-design the experiment to

solely measure the effect of different insertion points.

Exercises:

1) The students will identify the three levers involved in the action mimicking a free throw. The

motion of the arm is composed of three different levers: 2 third-class levers and 1 first-class.

(Note: You do not need to give them the hint of the different classes if you are confident

they can deduce them.)

i. 1st class: Extension of the triceps: load-ball in hand, fulcrum- elbow joint,

effort-triceps

ii. 3rd

class: Load-ball in hand, effort-muscle insertion holes on forearm,

fulcrum-elbow joint

iii. 3rd

class: Load-ball in hand, effort-wrist muscles insert distal to wrist joint,

fulcrum-wrist joint.

b. In groups of 2-3, the teacher should provide each group some kind of ball, like a

basketball to practice the free-throw motion.

c. The students need to work together looking at anatomy books and feeling their own

muscles while mimicking the action. Encourage the students to study the different

muscles and joints involved to determine the levers involved.

d. When the students have successfully identified the three levers have them draw the

isolated levers on their activity sheet labeling which muscles and joints are the

fulcrum, input and output forces.

NOVATeachers (2004).

12

2) Studying the mechanical advantage of the lever involving the forearm muscles.

a. Question: Which of the three levers are affected if we change the insertion point of

the muscles representing the triceps? (answer: the 3rd

class lever action of the

forearm: fulcrum=elbow, input = muscle insertion, load =ball.)

b. Have the students predict which muscle insertion is going to produce the greatest

distance thrown and why. Encourage them to think about 3rd

class levers and the

components of mechanical advantage.

c. After the prediction have each group perform five trials at each insertion point,

measuring the distance traveled. Make sure the students pull back the model the

same amount for each trial. The easiest release point is when the radius and ulna are

parallel to the table. Have the students record the data on the data table and find the

average for each insertion point.

d. Compare their results to the prediction and have them explain why the third insertion

point produced (hopefully) the furthest distance in terms of levers and mechanical

advantage.

e. You could also have the students measure the input and output distances to calculate

the mechanical advantage of each insertion point.

3) Identifying the scientific method fallacy in this activity.

a. Have the students identify the manipulated, controlled and responding variable(s) in

this experiment.

b. Is there anything wrong with this picture? (if the students do not identify that the

tension of the rubber band also is manipulated in addition to changing the insertion

point nudge them in that direction) Hopefully the students will recognize that having

two manipulated variables is a violation of the scientific method.

c. Working in groups, have the students re-design the experiment so they are solely

measuring the effect of the different insertion points.

d. If time permits, allow the students to make the three different sets of muscles to make

the tension equal, have them form another prediction and run the data sets again

comparing these results to the first data sets.

4) Bioengineering Activity (optional)

a. Once the students have completed the exercises using the free-throw model introduce

students to the field of bioengineering by posing this unguided inquiry question:

i. Re-engineer this model to where it is a leg model and it kicks a ball.

13

Lever Mechanics Student Worksheet

Objective: Classify the lever actions of the free-throw model and determine the effect of

changing the effort distance.

Materials: Basketball(optional) Anatomy references

Extra string Rubber bands

Measuring tool (tape measure) Free-throw motion model

1. Studying your free throw motion

a. In groups of 2-3, each student will practice a free throw motion paying attention to

the muscles and joints involved.

b. You will discuss with your partners and use anatomy references to label the muscles

and joints.

2. Classifying levers

a. There are three levers working together to produce the free-throw motion

(discounting the effect of the shoulder). Using your own motion and observing the

model find the three levers and classify them as 1st, 2

nd, or 3

rd class.

b. Isolate the three levers and identify the fulcrum, effort and load. Draw and record

these on the data sheet.

3. Mechanical advantage of levers

a. Which of the three levers are affected if the insertion points on the radius/ulna are

changed?

b. Predict which muscle insertion is going to produce the greatest distance thrown and

why? Include references to lever class and mechanical advantage.

c. One student will act as the releaser for all trials. Make sure they always release the

ball from the same point. The easiest way is to release when the radius and ulna are

parallel to the ground/table.

d. Use the insertion point closest to the elbow joint first. One student will release the

ball and the others will calculate the distance traveled. Record.

e. Perform five trials and then find the average distance.

f. Repeat steps d-e using the second and third insertion points.

g. Measure effort and load distances for each insertion point and calculate the

mechanical advantage.

h. Compare your results to your prediction. Did the evidence support or refute your

hypothesis? Why or why not? Include discussion of effect of mechanical advantage

on distance thrown.

4. Scientific Method

a. Identify the manipulated, controlled and responding variables.

b. What are some problems with the activity you just completed?

c. As a group, re-design the experiment so it is solely measuring the effect of changing

insertion points.

d. Make your suggested changes and run the data sets again. You will have to make

your own data tables.

e. Compare these results to your previous data set.

5. Bioengineering(optional)

a. Re-engineer this model to where it is a leg model and it kicks the practice golf ball.

14

Data Sheet

Which of the three levers are affected if the insertion points of the triceps are

changed?___________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

Class of lever:_________________________________________________________

Muscles and joints involved:______________________________________________

Draw and label the fulcrum, effort, and load:

Class of lever:_________________________________________________________

Muscles and joints involved:______________________________________________

Draw and label the fulcrum, effort, and load:

Class of lever:_________________________________________________________

Muscles and joints involved:______________________________________________

Draw and label the fulcrum, effort, and load:

15

Hypothesis: Which muscle insertion is going to produce the greatest distance

thrown and why? Include references to lever class and mechanical

advantage.__________________________________________________________

__________________________________________________________________

__________________________________________________________________

Insertion Point 1 Insertion Point 2 Insertion Point 3

Trial 1

Trial 2

Trial 3

Trial 4

Trial 5

Average Distance

Mechanical

Advantage

Discussion: Identify manipulated, controlled, and responding variables

Manipulated:___________________________________________________

Controlled:____________________________________________________

Responding:___________________________________________________

How does this activity violate the scientific method?________________________

__________________________________________________________________

__________________________________________________________________

___________________________________________________

Compare your results to your prediction. Did the evidence support or refute your

hypothesis? Why or why not? Include discussion of effect of mechanical

advantage on distance thrown.__________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

On a separate sheet of paper, re-design this experiment so you are only solely

measuring the effect of the different insertion points. The sections that must be

included are: materials, methods, hypothesis, identification of variables, and data

tables. Run the experiment.

16

Finding the Velocity of a Projectile Using Vectors

Objective: Students will be able to construct the resultant velocity vector from the arm model

and use the parallelogram method to draw the resultant vectors.

Safety considerations: The students are working with a tension-filled model, thus it would be

advised that they wear proper eye protection.

Exercises:

1. Design a makeshift protractor: There are two different routes that can be taken to find the

release angle.

a. Obtain a large piece of cardboard.

i. Use small protractor in the bottom center of the cardboard and extend the

major angles with a meter stick (90, 75, 60, 45, 30, 15)

ii. Cut out the Cardboard in a half-dome shape

b. The second option (less accurate) is to have a student use a meter stick showing

the angle and comparing it to a small protractor to find the release angle.

2. Finding the resultant vector of the projectile (Groups of 4)

a. Each group member is going to have a particular role: releaser, angle determiner,

and two people in charge of the stopwatches, and measuring distance.

b. Releaser: Needs to make sure the arm is brought back the same distance for every

throw. The easiest way is to release the ball when the radius/ulna are parallel to

the table.

c. Angle determiner: Using whatever makeshift protractor method, this person

needs to use a meter stick to re-create the angle from the base of the humerus to

where the ball was released. Either using the small protractor or cardboard

protractor, record the release angle.

d. Stopwatch: These people need to start the stopwatch when the ball was released

and stop when the ball hits the floor. The actual time will then be the average

time between the two. It would be advised to have someone stand right next to

the release point, and the other closer to where the ball hits. They will then mark

where the ball hits and measure the distance from the model.

3. Finding the release height:

a. Place the model at the first release point.

b. Have one student launch the ball and the other students are closely observing at

what height the ball is released.

c. Have the students agree upon a spot and measure the height from the ground/table

to that point and record in the data table. Perform a total of three trials and

calculate the average.

d. Repeat the steps for the second and third release points.

4. Measured data:

a. Perform five trials at each of the release angles. Move the dowels at the base of

the model to either the 1st, 2nd, or 3rd inserts to change the angles.

b. The release angle of each trial will be recorded and then the average taken.

c. The average times for each trial will be recorded, and then the final average taken.

d. The distance for each trial will be recorded, and then the average taken.

17

e. Each student will then have to calculate the vertical velocity of the ball for each

trial using the equation h(t) = -4.9t2 + vyt + h0

f. Each student will also calculate the horizontal velocity using the equation vx=

distance/time

g. Repeat the steps for each release angle.

h. Note: For assistance with the mathematics visits the Appendices at the back

of this module.

5. Graphing the vectors:

a. Students/class will need to come up with a graphing scale for the horizontal and

vertical velocities.

b. Students will then graph the horizontal and vertical components.

c. Using the parallelogram method, the students will diagram the resultant vector

which is v0. Using a ruler and the known scale record Vo

d. Using a protractor the students will determine the angle and compare it to the

measured angle.

6. Discussion points:

a. Make sure you have the students predict which release angle is going to produce

the greatest velocity and distance. Why?

b. Have students identify the manipulated, controlled, and responding variables.

c. What is the ideal release angle for a projectile object like a cannon ball and why?

Use your data and vectors to come to this conclusion.

7. Extension Opportunity (optional)

a. For a fun competition, purchase some cheap basketball hoops that stick to the wall

(available at any variety or dollar store).

b. Set up the hoop(s) at a known height and tell the students each group has 10 shots

and the group that makes the most baskets gets a prize, extra credit, or whatever

you choose. You could set a minimum amount of baskets to get a passing grade

for the activity.

c. Provide the students with piles of rubber bands, string, and other materials to re-

engineer the model to their liking.

d. Have the students evaluate their data to find the ideal angle of release, distance

away from hoop, etc…beforehand so they do not waste any shots.

e. You may want to present the mathematics to find the maximum height of the

different release points they measured.

i. Use the equation h(t) = -4.9t2 + vyt + h0.

ii. Because the motion of the flight is a parabola the maximum height is

when the time is at its halfway point.

iii. For help with the mathematics, refer to the Appendices at the end of the

module.

18

Vector Activity Student Worksheet

Objective: Construct the horizontal and vertical components and use the parallelogram method

to find the resultant velocity vector.

Materials: Free-throw model Large piece of cardboard (1 per group)

Meter stick Protractor

Stopwatches (2 per group) Graph paper

Tape measure Ruler

A. Roles

o You will be in groups of 4, your teacher will inform whether you get to choose your

roles or they will be assigned.

Releaser: You are in charge of using the free-throw model, launching the ball.

It is important that you release the ball at the same point for each trial. The

easiest way is to release when the radius/ulna is parallel to the ground.

Angle Determiner: Using the protractor you built in section A keep a close

eye on the model and mark the release angle.

Timer(2): One person will stand near the model and the other near where the

ball will land. Both students will press START when the ball is released from

the hand and STOP when it hit the ground. The student by the landing will

mark where the ball hits and the timers will measure the distance from the

model.

B. Building the protractor

o Obtain a large piece of cardboard

o Place a small protractor on the bottom center of the cardboard and make marks of the

major angles (15, 30, 45, 60, 75, 90, 105, 120, 135, 150, and 165)

o Use a meter stick to extend the angles.

o Cut out the cardboard in a half-moon shape like the protractor.

C. Finding the release height

o Place the model at the first release point.

o Have one student launch the ball and the other students are closely observing at what

height the ball is released.

o Agree upon a spot and measure the height from the ground/table to that point and

record in the data table. Perform a total of three trials and calculate the average.

o Repeat the steps for the second and third release points.

D. Finding the resultant velocity vectors

o Predict which release point is going to produce the greatest velocity upon release.

o Place the wooden dowels on the base in the first hole.

o The releaser will pull back the model and release the ball, angle determiner marking

the angle, and timers find the time in the air and distance the ball traveled.

Record.

Calculate vertical velocity using the equation h(t) = -4.9t + vy. + h0

19

Calculate horizontal velocity using Vx= distance/time

o Perform five trials.

o Repeat the steps at the other two release angles.

E. Graphing the vectors

o Come up with a scaling method for the magnitude the vertical and horizontal

components.

o Use ruler to draw the horizontal and vertical components

o Draw parallelograms to find the resultant vector.

o Use a ruler to determine the velocity of the resultant vector.

o Use a protractor to measure the release angle.

o Compare the graphed release angle to the measured angle.

20

Finding the Velocity of a Projectile Using Vectors

State your hypothesis: Which release point is going to produce the highest velocity

and why? ______________________________________________

__________________________________________________________________

________________________________________________________

Release

point 1

Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average

Release

Height (h0)

n/a n/a

Angle of

Release

Time

(sec)

Distance

(m)

Velocityy

(m/s)

Velocityx

(m/s)

Velocity0 n/a n/a n/a n/a n/a

Release

point 2

Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average

Release

Height (h0)

n/a n/a

Angle of

Release

Time

(sec)

Distance

(m)

Velocityy

(m/s)

Velocityx

(m/s)

Velocity0 n/a n/a n/a n/a n/a

21

Release

point 3

Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average

Release

Height (h0)

n/a n/a

Angle of

Release

Time

(sec)

Distance

(m)

Velocityy

(m/s)

Velocityx

(m/s)

Velocity0 n/a n/a n/a n/a n/a

Release point 1 Release point 2 Release point 3

Measured angle

Discussion:

Identify the controlled, manipulated, and responding variables in this activity:

Controlled:____________________________________________________

Manipulated:___________________________________________________

Responding:___________________________________________________

Compare your measured angle and the angle of release using the makeshift

protractor. Were they reasonably close? What are some factors that contributed to

their differences? ____________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

22

Compare your v0 values between the three release points. Did the data support or

refute your hypothesis? Why or why not? _______________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

What is the ideal release angle for a projectile object like a cannon ball and why?

Use your data and vectors to come to this conclusion._______________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

When working with this model, do you think that it represents an accurate

portrayal of a human arm? Why or why not?______________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

23

Uniform Motion Activity Materials

1. Stopwatch

2. Calculator

3. Model

4. Tape measure

Mathematics

1. , where t is the time measured in seconds, is the

vertical component ( velocity ) and is the initial height.

2. , where is the horizontal component (velocity) , t is the elapsed time(the time

the ball is in the air), and d is the distance the ball travelled.

3. , where is the velocity of the ball.

4.

5. , or

Students will use the model to make predictions as to the maximum height the ball will achieve

during its flight, and optimum angles that will achieve and furthest distance the ball will travel.

They will then check those results using the mathematics and hypothesize why or why not the

results vary with the initial predictions.

Step 1: Using the muscle insertion point in the first position, place the ball in the hand and

release the arm. Students will use the stopwatch to time the flight of the ball. Once the ball has

completed its flight, the students will measure the distance the ball has travelled.

Step 2: Once we have the time and distance we can calculate and . To find , substitute

the values t and found from the trials using the model into „1‟. Find using „2‟. Now, we

can use „3‟ to find the velocity of the ball.

Step 3: We are now ready to find the maximum height of the ball. Using the average time of the

flight of the ball, , substitute this number in for t in . The result will be the maximum

height the achieved during its flight.

Step 4: Calculate the angle, , that the ball was released using either the sine ratio or the cosine.

To do this, use either or the .

*(for an example, see the appendix)

24

Student Activity Worksheet

Uniform Motion Materials

5. Stopwatch

6. Calculator

7. Model

8. Tape measure

Mathematics

6. , where t is the time measured in seconds, is the

vertical component ( velocity ) and is the initial height.

7. , where is the horizontal component (velocity) , t is the elapsed time(the

time the ball is in the air), and d is the distance the ball travelled.

8. , where is the velocity of the ball.

9.

10. , or

1) With arm cocked and ready to launch, measure the distance from the ball to the floor and

record the result. This is the value in „1 „ above.

2) Using the model, a stopwatch and a tape measure, launch the ball three times and record the

time and distance in the table provided. Start the stopwatch upon release of the arm and stop

when the ball strikes the ground.

Trial 1 Trial 2 Trial 3 Avg. Trial

Time

Distance

3) Using the data you recorded, answer the following questions:

1) Find and . ( Hint: To find , set )

2) Using and , find . What does this quantity represent?

3) Calculate the maximum height the ball achieved.

4) Find the angle that ball was released at.

25

4) Repeat the above exercises by moving your stop block to the second and third locations.

Record all data.

Stop 2:

Trial 1 Trial 2 Trial 3 Avg. Trial

Time

Distance

Stop 3:

Trial 1 Trial 2 Trial 3 Avg. Trial

Time

Distance

5) What happens to the trajectory when the arm is elevated? Why?

6) At what angle do you think you would achieve a maximum trajectory? Why?

7) When working with this model, do you think that it represents an accurate portrayal of a

human arm? Why or why not?

8) If you answered no to question 7, what do you think you could do to the model to create

something that more closely resembles human motion?

26

Extensions Uniform Motion

If a CBR (Calculator Based Ranger, Texas Instruments) is available, it can be used by the

students to analyze uniform motion and interpreting data from graphs. This can be achieved by

having the students break up into small groups and have one student hold the CBR in front of the

model. Cock and release the arm and follow the instructions with the CBR to gather the data

from the motion of the ball. They can then interpret the data displayed from the graph(s) and

compare those to the data they gathered from the activities.

Transforming Arm Model to a Leg Model

Once the students have completed the exercises using the free-throw model introduce

students to the field of bioengineering by posing this unguided inquiry question:

o Re-engineer this model to where it is a leg model and it kicks a ball.

Free Throw Competition

For a fun competition, purchase some cheap basketball hoops that stick to the wall

(available at any variety or dollar store).

Set up the hoop(s) at a known height and tell the students each group has 10 shots and the

group that makes the most baskets gets a prize, extra credit, or whatever you choose.

You could set a minimum amount of baskets to get a passing grade for the activity.

Provide the students with piles of rubber bands, string, and other materials to re-engineer

the model to their liking.

Have the students evaluate their data to find the ideal angle of release, distance away

from hoop, etc…beforehand so they do not waste any shots.

You may want to present the mathematics to find the maximum height of the different

release points they measured.

o Use the equation h(t) = -9.8t2 + vyt + h0.

o Because the motion of the flight is a parabola the maximum height is when the

time is at its halfway point.

o For help with the mathematics, refer to the Appendices at the end of the module.

27

Appendix

This appendix contains all of the key mathematical concepts needed by the student to be

successful when attempting this module. Student exercises are include, with answers provided at

the end of the appendix.

Quadratic Formula

28

The Pythagorean Theorem

A

c

b

C a B

29

Right Triangle Trigonometry

A

c

b

C

a B

30

Uniform Motion

31

Vector Algebra When studying the magnitude of motion it is important to distinguish between distance and

displacement. Distance is the length of a path between

two points. It is important to remember that the path

traveled is an integral component of measuring

distance. In contrast displacement is simply the length

of a straight line between the starting and end point.

The figure below highlights this difference. The yellow

path shows the distance traveled by an individual taking

into consideration the particular route chosen. However

the green arrow shows the actual displacement between

the starting and ending points.

Displacement is an example of a vector. A vector is a

quantity that has both a magnitude and a direction. The

magnitude can be expressed in different ways such as

amount, length, size, etc… Vectors are written as

arrows and the length portrays the magnitude.

Vector addition is the combination of the magnitude

and directions therefore you can add displacements.

If the displacements are in the same direction you add

the magnitudes.

Ex:

+

=

When adding two vectors of opposite direction, subtract their magnitudes.

Ex: +

=

If displacement is not along a straight path they may be combined by using graphs. The single

vector that is obtained from the composition of two vectors is known as the resultant vector.

Ex: +

=

Note: = the resultant vector.

If two component vectors are at right angles, the resultant vector (hypotenuse) can be calculated

using the Pythagorean theorem (see Appendix on using this theorem).

Elert, Glen (1998).

32

The “tip to tail” method is used to derive the magnitude of a resultant vector. As shown in the

following figure the tip of vector „A‟ is connected

to the tail of vector „B.‟ The resultant vector

“A+B” is the vector from the tail of „A‟ to the tip

of „B.‟ It is important to note that vectors A and B

could be reversed and the magnitude would be

same.

Another method to add component vectors to

find the resultant is known as the

parallelogram method. In this method the

component vectors are connected to one

another and a parallelogram is traced as

shown in the figure to the right. The resultant

vector (R) is from the origin of AB to the

opposing corner of the parallelogram.

Sparknotes (2006).

Delpierre, GR & Sewell BT (1992)

33

Exercises:

1. Draw the resultant vector, when composing the following vectors:

a. + =

b. + =

c. + =

d. +

2. Compute the magnitude R of the resultant vector, given the magnitudes of vectors A and

B.

a. A = 3, B = 4, R = ? (A & B are in the same direction)

R

b. B

A

3. Draw the component vectors and determine the magnitude of the resultant vector.

a. A girl walks home from school to home the same way each day. She starts by

walking 2 blocks east, then turns a corner and walks one block north. She turns

once again and walks one block east. She finishes the walk home by going three

blocks to the north. What are the distance the girl traveled and the displacement

from the school to the house?

34

References

Barton, ML & Heidema, C (2002). Teaching and Reading in Mathematics (2nd

Ed.) MCREL.

Bioengineering Overview (2003). Sloan Career Center. Retrieved July 30, 2008 from

http://www.careercornerstone.org/pdf/bioeng/bioeng.pdf

Bionewsonline (2005) Transgalactic LTD. Retrieved July 30, 2008, from

http://www.bionewsonline.com/k/what_is_bioengineering.htm

Delpierre, GR & Sewell BT (1992). Electronic Science Tutor. Found July 30, 2008 from

http://www.physchem.co.za/Vectors/Addition.htm#Parallelogram

Elert, Glen (1998) Physics Hypertextbook. Found July 9, 2008 at

http://hypertextbook.com/physics/mechanics/displacement/

Secrets of Lost Empires—Pharoahs Obselisk (2004). NOVA Teachers. Found July 30, 2008

from http://www.pbs.org/wgbh/nova/teachers/activities/27po_sle2phar.html

Vector Addition (2006). Sparknotes. Found July 30, 2008 from

http://www.sparknotes.com/testprep/books/sat2/physics/chapter4section2.rhtml