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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: • • • • • • • 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 – – – – 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 – – • 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 – – Abilities of technological design Understandings about science and technology Mathematics Standards: 9-12 – – – – – – – – – – – – 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 • 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. • 3. Students use data collection and analysis, statistics*, and • probability* in problem-solving situations and communicate the reasoning used in solving these problems. • 4. Students use geometric concepts, properties, and relationships in problem-solving situations and communicate the reasoning used in solving these problems. • 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. • 6. Students link concepts and procedures as they develop and use computational techniques, including estimation, mental • arithmetic*, paper-and-pencil, calculators, and computers, in • problem-solving situations and communicate the reasoning used in solving these problems. SCIENCE • 1. Students understand the processes of scientific investigation • and design, conduct, communicate about, and evaluate such • investigations. • 2. Physical Science: Students know and understand common • properties, forms, and changes in matter and energy. • 5. Students know and understand interrelationships among • science, technology, and human activity and how they can • affect the world. • 6. Students understand that science involves a particular way of knowing and understand common connections among • 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)