Transcript Student Materials - Computational science

Module Description
This module incorporates 2-dimensional motion equations and applies them
to the real world phenomena of projectiles. Students will build and use
small-scale catapults to investigate the scenario of delivering supplies to a
group of hikers stuck across a deep river gorge, inaccessible by other
means.
In this module, computer modeling techniques using Excel and STELLA®
will be introduced to enable students to predict how far a projectile will travel
along the ground and how high it will rise when shot with a known initial
speed and given launch angle.
The module was designed to be used in Physics, and various Mathematics
classes. The timeline given is for a modified block schedule, but could
easily be adapted to any class schedule.
Objectives
At the end of this module students will be able to:
 Relate the height, time in the air, and initial velocity of a projectile
using its vertical motion and determine the range for a catapult.
 Predict the path of a projectile based on analysis of algebraic
equations.
 Use Excel and STELLA to model and interpret the path of a
projectile in 2-dimensional motion.
 Compare their experimental data with model curves and provide
explanations for any differences observed.
Requirements
Lab equipment:
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Timers
Meter-sticks
Calculators
Rubber bands
Popsicle sticks
Glue
Marshmallows
Computer Equipment:
• Personal computers with Microsoft Excel and Internet Capabilities
Content Knowledge:
• Students should be comfortable with the following concepts
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Velocity
Gravity
Acceleration
Cosine and Sine functions
Background Information #1
PHYSICS
• There are 2 components of velocity of any projectile, the horizontal
velocity (vx) and the vertical velocity (vy). These two components of
velocity are independent of each other so we may consider them
separately when developing our equations.
• Horizontal Component:
Once a projectile leaves the object that is launching it, there
are no forces acting on the projectile (neglecting air resistance).
Using Newton’s Laws, we know that an object in motion will
continue in motion unless acted upon by a force. Knowing this
we can say that the velocity in the horizontal direction is
constant. Therefore, we can use the simple velocity equation to
solve for the distance traveled in the horizontal direction by the
projectile:
X=vxi*t
Background Information #2
PHYSICS
• Vertical Velocity Component
When analyzing the vertical component of the velocity we must
remember that there is a force acting on the projectile. That force is
gravity. At any point on earth, neglecting air resistance, two balls
dropped at the same time will experience the same gravitational
acceleration and will therefore land at the same time. Remember, this
motion does not depend on the mass of the object! Again, neglecting
air resistance and combining equations for acceleration and velocity, we
come up with the equation for free-fall:
Y= ½ *g*t2
If an object is given an initial velocity in the vertical direction
(rather than simply being dropped), the equation becomes:
Y=vyi*t- ½ *g*t2
Because of the downward acceleration of gravity, the velocity decreases
until it reaches its highest point, at which point the velocity increases as
the projectile falls downward.
Background Information #3
PHYSICS
• In general, the overall initial velocity is measured
rather than measuring the initial velocity in the x
and y directions separately. To resolve the initial
velocity into it’s x and y components, we must
use trignometric relationships to develop the
following equations.
Vxi=cosθ*vi
Vyi=sinθ*vi
vi
θ
vyi
vxi
Definition of Variables
• Physics Background
 Vi: initial velocity
 Vxi: x-component of initial velocity
 Vyi: y-component of initial velocity
 g: acceleration due to gravity (-9.81 m/s2)
 t: elapsed time
 Θ: launch angle
References and Resources
• Applets and other Resources showing
projectile motion
http://www.phy.ntnu.edu.tw/java/projectile/projectile.html
http://www.phys.virginia.edu/classes/109N/more_stuff/Applets/Pro
jectileMotion/jarapplet.html
http://library.thinkquest.org/2779/History.html
http://www.sciencejoywagon.com/physicszone/lesson/01motion/p
rojecti/default.htm
• References
STELLA model adapted from a Maryland Virtual High School
Projectile Motion Stella Model.
http://www.ncsec.org/SC2001tm.cfm
STELLA software developed by High Performance Systems, Inc.
Lesson Plans
Day 1: Introduction to problem, identification of factors
Day 2: Pre-test, continue identification of variables
Day 3-5: Build and Test Catapults
Day 6: Group and Class Development of Flow Charts to Describe
Projectile Motion
Day 7: Introduction of 2-D motion equations (quadratic equation) and
practice
Day 8: Model development for Excel
Day 9: Model application in Excel
Day 10: STELLA model for scenario predictions
Day 11: Student presentations of scenario solutions
Teaching Tips
This laboratory exercise is designed to be able to be
performed in any classroom. If photogates and graphing
calculators are available, the lab can be made to include
use of this equipment. Without these materials (as we do
not have them), students will need to be more creative in
determining vi. In our classroom, we discuss what “initial”
means and then discuss possible ways of measuring this
velocity. Students take the first 10% of the projectile’s
path and record the time, the distance in the x-direction
and the distance in the y-direction to determine the initial
velocity.
In addition, if your school owns equipment for launching
projectiles, use of these items rather than having students
produce their own would allow for a shorter module.
Assessment Strategies
• Pre and Post Test on Overall Concepts
• Catapult Lab Grading Rubric with Peer
Evaluation
• Group presentation and class evaluation of
models
• Assessment of Excel Spreadsheet and Graphs
with Graphing Rubric
• Final Assessment of Student Presentations
– Students choose oral presentation, written report,
powerpoint presentation or other method of their
choice
National Standards
Science Content Standards: 9-12
• CONTENT STANDARD A: As a result of activities in grades 9-12, all students
should develop
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Abilities necessary to do scientific inquiry
Understandings about scientific inquiry
CONTENT STANDARD E: As a result of activities in grades 9-12, all students
should develop
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Abilities of technological design
Understandings about science and technology
Mathematics Standards: 9-12
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Mathematics as Problem Solving
Mathematics as Communication
Mathematics as Reasoning
Mathematical Connections
Algebra
Functions
Geometry from a Synthetic Perspective
Geometry from an Algebraic Perspective
Trigonometry
Discrete Mathematics
Conceptual Underpinnings of Calculus
Mathematical Structure
Colorado Standards
MATHEMATICS
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2. Students use algebraic methods* to explore, model*, and describe patterns* and functions*
involving numbers, shapes, data, and graphs in problem-solving situations and communicate
the reasoning used in solving these problems.
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3. Students use data collection and analysis, statistics*, and
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probability* in problem-solving situations and communicate the reasoning used in solving these
problems.
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4. Students use geometric concepts, properties, and relationships in problem-solving situations
and communicate the reasoning used in solving these problems.
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5. Students use a variety of tools and techniques to measure, apply the results in problemsolving situations, and communicate the reasoning used in solving these problems.
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6. Students link concepts and procedures as they develop and use computational techniques,
including estimation, mental
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arithmetic*, paper-and-pencil, calculators, and computers, in
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problem-solving situations and communicate the reasoning used in solving these problems.
SCIENCE
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1. Students understand the processes of scientific investigation
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and design, conduct, communicate about, and evaluate such
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investigations.
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2. Physical Science: Students know and understand common
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properties, forms, and changes in matter and energy.
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5. Students know and understand interrelationships among
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science, technology, and human activity and how they can
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affect the world.
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6. Students understand that science involves a particular way of knowing and understand
common connections among
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scientific disciplines.
Cross-Curricular Integration
• This module can be integrated into
instruction of several different
departments.
– English: Beowulf
– PE: Sports strategies and techniques
– History: Medieval Study
Glossary
Projectile – An object with independent vertical and
horizontal motions that moves through the air only
under the force of gravity after an initial thrust
Trajectory – The path of a projectile through space
Range – horizontal distance the projectile travels
Flight Time – time the projectile is in the air (also
called hang time in sports)