CHANGE: THE PATH OF A THROWN BALL
Grades 8-12

When an object is thrown, shot, or otherwise propelled into the air, it will follow a path through the air known as a trajectory. If no forces other than gravity act on the object, its path can be easily and accurately determined. In this physics or physical science lesson, students in grades 9-12 will learn to apply Newton's laws and kinematic equations to predict a projectile's path. If the mathematical equations are omitted, the lesson can be adapted for grade 8. In the video, objects moving under conditions of constant velocity and constant acceleration are clearly demonstrated. The hands-on activity provides students with an opportunity to construct a physical model to test their prediction. The investigation can be completed in two to three class periods.
This lesson can serve as a prerequisite for an accompanying lesson entitled Curve Balls in which students will use a qualitative understanding of Newton's laws to explain how real variables such as air resistance and spin will alter the predicted path of a projectile.

3-2-1 Classroom Contact #20: Play Ball!

Students will be able to:

• understand Newton's laws as they apply to the motion of a projectile under ideal (i.e. without air resistance) conditions.
• draw force diagrams of a projectile under
• ideal conditions.
• apply basic kinematic equations to the motion of
• a projectile.
• describe, draw, and calculate the approximate path of a projectile under ideal conditions.

Texas Assessment of Academic Skills (TAAS), Grade 8
Science Objectives:
#3: Communicate scientific data and/or information.
#4: Interpret scientific data and/or information.
#5: Make inferences, form generalized statements and/or make predictions.
#7: Draw conclusions about the process(es) and/or outcome(s) of a scientific investigation.
#8: Relate and apply scientific and technological information to daily life.
Math Objectives:
#11: Determine solution strategies and analyze or solve problems.
#12: Express or solve problems using mathematical representation.
#13: Evaluate the reasonableness of a solution to a problem situation.

NCTM Standards for Grades 9-12
Standard 13: Conceptual Underpinnings of Calculus
Standard 14: Mathematical Structure

Per group of 3-4 students:

• 4 five inch rings and ring stands (rings should be adjustable vertically and able to rotate)
• 2 meter sticks
• projectile launcher, round projectile, and photogate timers, or
• 1 inch diameter metal ball, grooved ramp, and stop watch
• assignment sheet (included with this lesson)
• Shooting Hoops activity sheet (included with this lesson)

• projectile-object moving under the influences of inertia and gravity.
• trajectory-the path a projectile traces as it moves.
• vector-a physical quantity possessing a magnitude and direction.
• component vectors-two or more vectors which can be combined mathematically to produce a resultant vector.
• range-distance a projectile travels in horizontal direction.
• kinematics-the study of objects in motion.
• independence of motion-the phenomena in which the horizontal motion of a projectile appears to act independently of the vertical motion, (i.e. object moves horizontally with constant velocity and moves vertically with constant acceleration). This apparent independence allows the motions to be separated and calculated independently of one another.

As an assignment the evening before this lesson, students should review Newton's first and second laws, the corresponding force diagrams, and kinematics equations for calculating velocities and distances. Students should be able to calculate velocities and distances of objects moving under conditions of constant velocity and constant acceleration (most notably, acceleration due to gravity). Students should be able to draw appropriate force diagrams for each situation and apply Newton's laws appropriately. Students should also be able to combine horizontal and vertical velocity components using the Pythagorean Theorem. These concepts will be reviewed briefly before viewing the video.
Ask a student to state Newton's first law. After it has been properly stated, put it on the board or the overhead. Ask another student to draw a force diagram corresponding to the first law. Ask if anyone can draw a different force diagram which still indicates constant velocity motion. Diagrams should indicate no forces or balanced forces. Ask a student to provide an equation for calculating the distance travelled by an object under first law conditions. (d = v*t)
Ask a student to state Newton's second law. After it has been properly stated, put it on the board or the overhead. Ask another student to draw a force diagram corresponding to the second law. Ask if anyone can draw a different force diagram which indicates constantly accelerated motion. Diagrams should indicate unbalanced forces. Ask a student to provide equations for velocity and distance travelled by an object travelling under second law conditions. (v = v0 + a*t ; d = v0*t + 1/2 * a*t2)
Diagram an object moving with a constant horizontal velocity of 10 m/s and a constant vertical velocity of 5 m/s. Ask one student to indicate the direction of resultant velocity, a second student to calculate resultant velocity, and a third to calculate the distance it will travel in three seconds. Indicate that the path the object follows is called its "trajectory". If the object is given an initial velocity and then allowed to move on its own, it is called a "projectile". Ask students if all objects moving in two dimensions always move in constant velocity conditions.

To give the students specific responsibilities while viewing, ask students to watch the video and be prepared to:
a) draw force diagrams and describe motion of specific objects in the video; and
b) qualitatively predict the two dimensional path of specific objects in the video and support predictions based on Newton's Laws and kinematic equations.

BEGIN the 3-2-1 Classroom Contact video without sound when the host is observing the motionless ball. PAUSE the tape. Have students draw a force diagram of the ball, describe the motion, and indicate which of Newton's laws corresponds to the situation. After an appropriate time, ask students to respond individually or as a group.
FAST FORWARD the video to the point where the host has the ball hanging from a string. RESUME the video showing the ball hanging motionless on the string. PAUSE the video. Have students draw a force diagram of the ball, describe the motion, and indicate which of Newton's laws corresponds to the situation. After an appropriate time, ask students to respond individually or as a group.
RESUME the video. PAUSE the video when the young man kicks the ball. Have students draw a force diagram of the ball, describe the motion, and indicate which of Newton's laws corresponds to the situation. After an appropriate time, ask students to respond individually or as a group.
Softly kick a soccer or basketball across the floor. Ask the students to describe the motion of the ball after it leaves your foot. Students should understand that after leaving your foot, the ball travels at approximately constant velocity, with some loss due to friction. Students should clearly note that the ball does not speed up after leaving your foot. Provide a reasonable distance and time. Ask students to calculate the average velocity of the ball.
RESUME the video. PAUSE the video just as the host produces the scissors. Ask students to predict how she plans to make the ball move. RESUME the video to see if the prediction is right. PAUSE the video just as she cuts the string. Have students draw a force diagram of the ball, predict the motion, and indicate which of Newton's laws corresponds to the situation. After an appropriate time, ask students to respond individually or as a group.
Ask how this situation is different than kicking the ball. Students should indicate that this is a situation of constant acceleration due to a constant force and gravity, and the ball continues to speed up as long as it is falling. Provide a reasonable height. Ask students to calculate the time it will take the ball to reach the ground. Go through the calculation on the board.
RESUME the video. PAUSE just before the pitching machine throws the ball. Have students form groups of 3-4. Ask students to draw a force diagram of the ball, predict the path of the thrown ball, and explain the reasoning based on Newton's laws and kinematics equations. After three to four minutes, ask a group to explain its prediction and draw its path on the board. Ask other groups if they support the prediction and have them explain why or why not. [Note: Projecting graph paper on the via an overhead projector may help students sketch the path more accurately.] Students should ultimately conclude that the ball must move with constant velocity horizontally, with constant acceleration vertically, and with a parabolic path. The path starts out nearly horizontal but curves downward at an ever steepening angle throughout the flight.
RESUME the video. PAUSE the video between each flight of the ball and ask students if the actual path is consistent with the prediction. Ask students why it is important (in terms of the demonstration) for the pitching machine to give the ball a consistent initial velocity and height. Students should recognize these as experimental controls which allow easy comparison of trajectories. Ask students to predict how flight will change if the pitching machine (a) applies a greater initial velocity, (b) applies a smaller initial velocity, (c) is raised up another 12 inches. Have students draw the different force diagrams and predicted paths in their notebooks. Have groups draw and explain different situations on the board.
RESUME the video. View the segment of video showing the ball being shot through hoops. STOP the video after the ball has gone through the hoops once. Ask students to determine how the positioning of hoops could be predicted without using trial and error.

After viewing the video, drawing the diagrams, and answering the questions, the students should have a good qualitative feel for the trajectory of a projectile under the influence of gravity. They should now be able to apply the kinematic equations to calculate the height of a projected ball at various distances from the point of projections. Providing several worked examples and homework problems will help to reinforce these calculations. In the following activity, students will experimentally test their understanding of the motion of a projectile.
For the remainder of the period, groups of three or four students should discuss how they could predict the positioning of four hoops so a horizontally projected ball would travel through unobstructed. Stop the groups with five minutes left to review what they have learned about projectile motion. Emphasize the following: (a) horizontal motion is at a constant velocity throughout the path; (b) vertical motion is at constant acceleration; (c) the initial vertical velocity of a horizontally thrown object is zero; and (d) the kinematics equations they already know are valid for projectiles.
Assignment: Students should complete the following homework assignment:
1) Determine what variables you would need to know to determine the height of a ball at any given distance from the point of projection and a method for experimentally determining those variables.
2) On the assignment sheet (included with this lesson), calculate the range, height, horizontal and vertical velocities of horizontally projected objects using kinematics equations.

On the following day, briefly review the concepts/solutions to the homework problems. Students must clearly understand that, during the flight, the ball travels with constant horizontal velocity and constant vertical acceleration. Students must be able to calculate the time of flight and the vertical position of a projectile, given the initial horizontal velocity of the ball. Students must also be able to calculate an average velocity given the distance and time of travel.
In the following activity, students will predict the path of a horizontally projected ball by calculating the height of the ball at several horizontal positions. They will test the predicted path by placing rings at the appropriate heights to allow the ball to travel through the rings. The students should neither guess at the position of the rings nor place the rings using trial and error. Instead, the students must use their knowledge of an object's motion and the kinematics equations to determine the appropriate heights.

Activity: Shooting Hoops
A method must be developed to project a ball horizontally at a consistent and measurable speed. It may be worthwhile to allow students to determine a method. One route is to use a projectile launcher with photogates either purchased from a laboratory equipment company or borrowed from a local university. If this is not possible, a steel ball may be accelerated from a specified height down an incline plane, rolled across a flat smooth table, and projected off the end. The velocity of the ball may be calculated by timing the ball as it travels between two marked points on the flat section. Averaging the times of several runs will help reduce error. It is important to stop the ball before it leaves the table to eliminate the opportunity to watch the ball's trajectory before calculating it. Making a grooved ramp and track with pieces of 2x4 may reduce aiming problems significantly.
Distribute the Shooting Hoops activity sheet (included with this lesson) to each group. Provide sufficient time and space for groups to conduct the activity. After the activity is completed, allow time for students to discuss what they have learned and to clarify concepts needed for the experiment report. The following questions can be included: How accurate were the predictions? What are possible sources of error in the experiment? (friction, poor timing of the ball in Step One, aim, etc.) Was the path of the ball consistent with what you have learned about projectile motion? If the path was consistent, what if anything does that say about the validity of Newton's laws and the kinematic equations?

Have pairs of students find a building of some height off which a ball can be safely thrown. Taking turns throwing the ball, the students watch the ball's path to the ground. They sketch its path, calculate its range and time of flight, and compare to real values. Have them find out how to calculate the range of balls hit upward at an angle. Calculate the initial velocities needed to hit home runs, kick field goals, drive a golf ball from a tee to a green, etc.

If the baseball team has two pitching machines, ask the coach to set the machines at different initial velocities. Compare and contrast the paths of the balls. Note especially which ball hits the ground first, which ball travels the farthest, and which ball has the flattest path.

History: Students can research the evolution of pitching throughout baseball, as well as the evolution of baseball in American society. The Ken Burns' series, Baseball is an excellent resource. An especially interesting segment in Baseball's Program #7 shows Willie Mays' famous over the shoulder catch and throw to the plate, considered by many the best fielding play in the history of the game.
History: Students can study the development of artillery such as catapults, cannons, and airplane bombs, which rely heavily on the ability to calculate trajectories. An especially interesting example is Germany's development of Big Bertha, a long range artillery gun which was a failure due to the problems correctly calculating the path of the shell.
Language Arts: Students can read a variety of books about baseball and its role in American life. One especially good book is W.P. Kinsella's Shoeless Joe, an exploration of the mystical effect of baseball on the American pysche. It is the book upon which the movie "Field of Dreams" was based.
Math: The calculation of the path of the ball in the hoops activity provides an excellent application of graphing skills and work with quadratic equations.

1995-1996 National Teacher Training Institute / Austin

Master Teacher: Brendan Maxcy