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
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