Biomechanics of Free-Throw Motion—Uniform Motion and …



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.

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

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.

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).

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]

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

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.

[pic]

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

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.

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.

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.

a. In groups of 2-3, the teacher should provide each group some kind of ball, like a basketball to practice the free-throw motion.

b. 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.

c. 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.

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.

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, 2nd, or 3rd 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.

Data Sheet

Which of the three levers are affected if the insertion points of the triceps are changed?_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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.

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.

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.

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

▪ 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.

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 | |

|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? ____________________________________________________

____________________________________________________________________________________________________________________________________

__________________________________________________________________

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?______________________________

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Uniform Motion Activity

Materials

1. Stopwatch

2. Calculator

3. Model

4. Tape measure

Mathematics

1. [pic], where t is the time measured in seconds, [pic]is the vertical component ( velocity ) and [pic]is the initial height.

2. [pic], where [pic] 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. [pic], where [pic] is the velocity of the ball.

4. [pic]

5. [pic], or [pic]

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 [pic] and [pic]. To find [pic], substitute the values t and [pic]found from the trials using the model into ‘1’. Find [pic] 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, [pic], substitute this number in for t in [pic]. The result will be the maximum height the achieved during its flight.

Step 4: Calculate the angle, [pic], that the ball was released using either the sine ratio or the cosine. To do this, use either [pic] or the [pic].

*(for an example, see the appendix)

Student Activity Worksheet

Uniform Motion

Materials

5. Stopwatch

6. Calculator

7. Model

8. Tape measure

Mathematics

6. [pic], where t is the time measured in seconds, [pic]is the vertical component ( velocity ) and [pic]is the initial height.

7. [pic], where [pic] 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. [pic], where [pic] is the velocity of the ball.

9. [pic]

10. [pic], or [pic]

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 [pic] 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 [pic] and [pic] . ( Hint: To find [pic] , set [pic])

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

3) Calculate the maximum height the ball achieved.

4) Find the angle [pic] that ball was released at.

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?

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.

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

[pic]

The Pythagorean Theorem

[pic]

[pic]

Right Triangle Trigonometry

[pic]

[pic]

Uniform Motion

[pic]

[pic]

[pic]

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).

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.

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?

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

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

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

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

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

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

-----------------------

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:

NOVATeachers (2004).

Elert, Glen (1998).

Sparknotes (2006).

Delpierre, GR & Sewell BT (1992)

A

c

b

C a B

A

c

b

C

a B

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download