Using Parametric Equations to Model Projectile Motion

by

Mary Eager

Objectives:
Students will model the motion of a projectile using parametric equations and will solve problems related to the motion of a projectile, its trajectory, and its range.

Materials: Textbook, Merril Advanced Mathematical Concepts, graphing caluclator, or other graphing utility for parametric equations, nerf ball

Lesson Description: The pitcher for the high school baseball team in the first period pre-calculus class motivates the class to take this lesson seriously and apply its concepts immediately. Students take turns tossing a nerf ball into a trash can. Conversation includes questions about 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.

Students read on page 449, "Objects that are launched are called projectiles. The path of a projectile is called its trajectory. The horizontal distance that a projectile travels is its range. Physicists describe the motion of a projectile in terms of it position, velocity and acceleration... As the ball moves, gravity will act on it in the vertical direction. The horizontal component will be unaffected by gravity. So, discounting wind resistance, the horizontal speed of the ball is constant throughout the flight of the ball. The vertical speed is large and positive at the beginning, decreasing to zero at the top of its trajectory, then increasing in the negative direction as it falls. " With having thrown projectiles for most of their sixteen to 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 some students and the mathematics describing projectile motion is new to most of the students.

In order to find the parametric equations that represent the path of the projectile, right triangle trigonometry is used to resolve the initial velocity into its horizontal and vertical components. 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 time is

x=t|v|cos( q)

where v is the initial velocity and q the angle, with the horizontal, at which the projectile is launched. The vertical position of the projectile after t time is the sum of the distance traveled due to the initial velocity and the distance traveled due to gravity. The height of an object affected by gravity is given by the equation

h=(1/2)g*t*t

where g=-9.8 meters per second squared or -32 feet per second squared according to the units of the problem. Using right triangle trigonometry and adding the effects of gravity the vertical position of the projectile is represented by the following equation:

where t is time and g is the acceleration due to gravity.

Equipped with these parametric equations students practice evaluating them for specific problems such as:

"Zoe kicked a soccer ball with an initial velocity of 45ft/sec at an angle of 32 degrees to the horizontal. After 0.6 seconds, how far has the ball traveled horizontally and vertically?"

The problem situation can be a little more complicated if the object is not launched from the ground. The vertical height at the time of launch must be added to the expression for y . The following problem illustrates this concept.

"Denise shoots and arrow with an initial velocity of 65 m/s at an angle of 4.5 degrees with the horizontal at a target 70 meters away. If Denise holds the bow 1.5 meters above the ground when she shoots the arrow, how far above the ground will the arrow be when it hits the target?"

Students write the parametric equations to model the path of the arrow. Then they find the amount of time that it willl take the arrow to travel 70 meters horizontally. This is when the arrow will hit the target. Using that value of time in the expression for y , they find the vertical position.

Graphing Calculator Activity : With the TI-82 in parametric mode it is possible to graph these equations to see the path of the projectile. One of the problems used in this section follows.

"On the sixth hole, Jack Nicklaus selects a five iron. He estimates the distance to the pin to be 200 yards. Nicklaus' swing provides an initial velocity of 150 ft/sec at an angle of 25 degress above the horizontal. Use a graphing calculator to determine if he will hit the pin." (Merrill, p452)

To begin students write parametric equations to describe the position of the ball after t seconds. Setting t min=0, t max=5, t step=.1, x min=0, x max=600 xscl=20, y min=-20, y max=50 and yscl=10 enable to students to view the graph of the equations and decide, "Did Jack hit the pin?" The point on the graph where y=0 is when the ball is on the ground. It is readily visible that the ball falls short of the pin. This is a good problem because it requires some unit conversions and thought process in order to set the viewing window to appropriate values. The analysis of the graph generates lots of discussion.

Assessment: Students analyze two problems and present their findings in writing. The problems are from page 453 of Merrill Advanced Mathematical Concepts.

"Grtechen Austgen, an outfielder for the West Chicago Wildcats, is 215 feet from home plate after catching a fly ball. The runner tags third and heads for home. Gretchen releases the ball at an initial velocity of 75 ft/sec at an angle of 25 degrees with the horizontal. Assume Gretchen releases the ball 5 feet above the ground and aims it directly in line with the plate.

a. Write two parametric equations that represent the path of the ball.

b. Use a calculator to graph the path of the ball. Sketch the graph shown on the screen.

c. How far will the ball travel horizontally before hitting the ground?

d. What is the maximum height of the trajectory?

e. The ball will fall short of reaching home plate. Could Gretchen change the initial angle in order for the throw to reach the plate? If so, find the angle she should use."

The other problem is, "An airplane flying at an altitude of 3500 feet is dropping supplies to researchers on an island. The path of the plane is parallel to the ground at the time the supplies are released and the plane is traveling at a speed of 300mph.

a. Write parametric equations that represent the path of the supplies.

b. Graph the path of the supplies on your graphing calculator and sketch the graph.

c. How long will it take for the supplies to reach the ground?

d. How far will the supplies travel horizontally before they land?"

Many other application problems can be found to use for practice and evaluation.