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![]() Using right triangle trigonometryand adding the effects of gravity the vertical position of the projectileis represented by the following equation: Where g=-9.8 meters per second squared or -32 feet per second squaredaccording to the units of the problem. The height of an object affected by gravity is given bythe equation The verticalposition of the projectile after t time is the sum of thedistance traveled due to the initial velocity and the distance traveleddue to gravity. Where v is the initial velocity and q theangle, with the horizontal, at which the projectile is launched. Since its unaffected by gravity,the horizontal speed will be the magnitude of the horizontal component.So the horizontal position of the projectile after t timeis ![]() In order to find the parametric equations that represent the path of theprojectile, right triangle trigonometry is used to resolve the initial velocityinto its horizontal and vertical components. " With having thrown projectiles for most of their sixteento eighteen years on this earth, these concepts are not new to the students.The language with which to describe their experiences might be new to somestudents and the mathematics describing projectile motion is new to mostof the students. The vertical speed is large and positive at the beginning, decreasingto zero at the top of its trajectory, then increasing in the negative directionas it falls. So, discounting wind resistance,the horizontal speed of the ball is constant throughout the flight of theball. The horizontalcomponent will be unaffected by gravity. As theball moves, gravity will act on it in the vertical direction. Physicists describe the motion ofa projectile in terms of it position, velocity and acceleration. The horizontal distancethat a projectile travels is its range. Students read on page 449, "Objects that are launched are called projectiles.The path of a projectile is called its trajectory. Conversation includes questionsabout how the speed and angle at which it is thrown help determine its path.How could the speed and angle be measured? Jeremy Grizzle, the pitcher,wants to know if there is an equation to tell the " most efficient"way to throw the ball. Studentstake turns tossing a nerf ball into a trash can. Lesson Description: The pitcher for the high schoolbaseball team in the first period pre-calculus class motivates the classto take this lesson seriously and apply its concepts immediately. Materials: Textbook, Merril Advanced MathematicalConcepts, graphing caluclator, or other graphing utility for parametricequations, nerf ball Objectives: Students will model the motion of a projectile usingparametric equations and will solve problems related to the motion of aprojectile, its trajectory, and its range. Using Parametric Equations to Model Projectile Motion
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