CHANGE: CURVE BALLS
Although the specifics of air resistance on a rotating object are complicated, the general changes in motion can be clearly explained and contrasted with the "ideal" conditions of zero air resistance. This physics or physical science lesson is a companion to the lesson entitled The Path of a Thrown Ball, which is an investigation where objects propelled into the air had no forces other than gravity acting upon the object. Here the discussion of a ball's trajectory is extended to include the effects of more complicated forces caused by air resistance. Through discussion and hands-on activity, students will practice the use of kinematics equations for falling and projected objects and apply knowledge of Newton's Laws to analyze the apparent forces on a moving ball under real and ideal conditions. This investigation can be completed in two to three class periods.
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.
3-2-1 Classroom Contact #20: Play Ball!
Students will be able to:
1. indicate the direction of an applied force by analyzing an object's motion.
2. draw force diagrams of a ball under various conditions based on an analysis on motion.
3. draw and contrast the paths of balls pitched under different conditions.
4. use and describe an apparatus to model real conditions.
Texas Assessment of Academic Skills (TAAS), Grade 8
#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.
#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:
- several small balloons
- 1 helium-filled balloon
- rubber bands
- paper clips
- aquarium (approx. 2ft x 1.5ft x 3ft) filled 3/4 full of water
- magnus force-force caused by differential air resistance on opposite sides of a spinning ball
- air resistance-resistive force on an object traveling through air
- ideal conditions-situation in which approximations are made in describing an experiment to eliminate complicated and difficult to control variables, such as air resistance
- real conditions-actual physical situation including all variables
Review the hoop activity from the companion lesson entitled, "The Path of a Thrown Ball." Clarify constantly accelerated vertical motion and constant velocity horizontal motion. Verify that students can draw force diagrams and interpret motion through Newton's Laws (inertia and F=ma) and kinematic equations (v = v0 + at, d=vavt, d = 1/2at2). Ask students to identify some of the assumptions made in calculating the path of the ball, such as no air resistance and constant force of gravity.
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 according to Newton's Laws; and
b) Qualitatively predict the two dimensional path of objects in the video and support predictions based on Newton's Laws and kinematic equations.
The first segment of the 3-2-1 Classroom Contact video reviews the hoop activity from the previous day. BEGIN the video with sound as the pitching machine is throwing the balls and the host is setting up hoops. Point out the consistent motion of the balls. PAUSE the video when host asks, "Why isn't baseball a boring game if the ball's motion is so consistent?" Ask students why the host thinks baseball could be boring. Why is it not easy to hit a ball? Why do the best major league players miss 7 out of 10 tries? Students should come up with several reasons. (left to right placement of ball, short time to hit ball, small bat and ball, changing initial velocity) Point out that according to calculations, the ball should follow a fairly regular path. Some students will know that the spin effects the ball. Ask them to explain how if they can.
RESUME the video. PAUSE the video when Professor Brant says, "The ball is going about 70 mph." Have students calculate the time to hit the ball if the mound is 60' 6" away from the plate.
RESUME the video. PAUSE the video when the pitcher winds up and says, "Here's the riseball." Ask students to explain, based on what they know about the ball's path, if the ball can "rise" on the way to the plate. Ask them to use Newton's Laws and draw a force diagram.
RESUME the video. PAUSE the video after you hear, "Some pitches seem to defy gravity. Some rise while some fall." Ask students if a ball can "defy" gravity. Students should understand that, since gravity is always present, the ball cannot defy gravity. Ask students how the ball might appear to defy gravity. Give students some hints by asking what assumptions were made in calculating the ball's path through the hoop. Would air resistance affect the path of the ball?
RESUME the video. PAUSE the video when professor Brant tells the host he had a half a second to hit the ball. Have students check calculations of time to hit the ball.
RESUME the video. PAUSE the video after you hear, "First it's important to understand how a ball moves without air resistance." Discuss this statement with students. Students should understand that the "ideal" condition of no air resistance will provide a point of comparison to the "real" conditions. It should be clear to students that the hoop activity on the previous day assumed ideal conditions. No adjustments were made to account for air resistance. In the video, the ball hitting the falling can is simply a variation of the hoop activity.
RESUME the video. PAUSE the video on the animation of the cannon before the ball is shot. Ask students to write a prediction of which will hit first and have them explain their reasoning. Take a poll of predictions. RESUME the video. PAUSE the video when the ball hits. Check the poll and ask a student to explain.
RESUME. PAUSE the video after the animated demonstration. Give students an arbitrary height of the cannon and ball and have them calculate the time to hit the ground. Emphasize that even though the cannon ball moves horizontally, its vertical motion is independent of that horizontal motion and is exactly the same as the falling object.
If students think it is a trick, it can be easily demonstrated in class. Attach two small boxes on either side of the end of a flexible ruler. Place a penny in each box. Holding the ruler horizontally, snap the end of the ruler so that one penny is shot forward while the other penny drops straight down. Ask students to listen and watch to determine which hits first. It should be almost exactly the same time.
RESUME the video. PAUSE when the host says, "Pitchers know a few tricks." Emphasize the predictable path of the ball due to the constant effect of gravity. Ask how pitchers can overcome this. Ask students to watch the video carefully and take notes on how the wind tunnel models real conditions.
RESUME the video to show the wind tunnel. PAUSE the video as the wind tunnel is turned on. Ask students the important features of the wind tunnel. (forced air, rough ball, ability to spin ball, balanced ball) Ask students why it is important to have a balanced ball. Students should understand that, because the force of gravity has such a dramatic effect on a falling ball, it is difficult to see other effects such as those of spin when a ball is thrown. The balanced ball allows us to view the effects of spin on a relatively stationary ball. Prompt students to note the spin of the ball, the direction of motion of the wind, and the ball's movement in the following video segment.
RESUME the video. PAUSE the video as the ball rises. Ask students to draw a force diagram of the ball and explain the motion using Newton's Laws. Have a student explain the force diagram. Tell students that the force of air resistance on a spinning ball is referred to as the "Magnus effect" or "Magnus force". Ask the class to predict what will happen to the ball if the spin is reversed.
RESUME the video. PAUSE the video when the ball sinks and Professor Brant says, "The ball sinks." Ask students if they predicted a sinking ball.
RESUME the video. Continue until the host gets a hit, have class cheer and do the "wave". STOP the video. [Note: It may be worthwhile to rewind and play the wind tunnel video segment a second time so that students can verify the spin direction and deflection.]
Discuss how the varying air resistance creates a force on the ball. The nature of this effect is discussed in detail in The Physics of Baseball by Robert Kemp Adair. The most important point is that air resistance increases with increasing velocity, so the side of the ball spinning into the wind has a greater force of resistance on it. Thus a fastball with back spin is forced upward slightly, while a curve with top spin will be forced downward. The best spin effect on a baseball is about one third the effect of gravity, so a "rising" fastball actually drops only slightly less than a normal pitch. Most of the deflection of the ball occurs in the second half on the ball's flight, making it even more difficult to detect.
Ask the students to draw force diagrams of the different spin situations, top and back, based on the motion of the ball in the wind tunnel. Ask them to predict the paths of the ball compared to a non-spinning ball. Ask them to predict the effects, draw force diagrams and describe the paths of balls thrown with side spin.
Activity: Modeling The Effect Of The Magnus Force On A Pitched Ball
Review with students that at any instant as a ball is moving through the air, there are a variety of forces acting on it. The force of gravity and forces arising from air resistance have the most measurable effects. Gravity forces the ball to accelerate downward dramatically during the flight. A non-spinning ball undergoes a significant drag force while in flight, reducing the ball speed by as much as 10 mph. As was shown in the video, a spinning ball is affected by differing drag forces (referred to as the Magnus force) on opposite sides of the ball, causing a deflection (change in direction).
Because gravity has such a dramatic effect on a thrown ball, it is difficult to see the more subtle effects of air resistance. The largest Magnus force on a very well thrown curveball is only one third the effect of gravity, and it acts in approximately the same direction, making the effect very difficult to see. In this activity, student will model the trajectory of a ball under the influence of only a small force such as the Magnus force. By controlling the greater effect of gravity, they can understand how these smaller forces alter the otherwise predictable path of a pitched ball.
Distribute the sheet entitled Activity: Modeling the Effect of the Magnus Force on a Pitched Ball (included with this lesson). Provide students with the materials, time and space to conduct the activity. Have students work in groups of three to four.
After groups conduct the activity, analysis should follow. The graphing exercises may be done qualitatively or quantitatively based on the level of sophistication of the students. In Analysis exercise #1, students should comprehend that the paths of balloons with upward and downward forces will have gradually increasing slopes. Students should also reason through how the extra forces can be combined with that of gravity to alter the motion of the ball. Force diagrams will be very helpful. Information from the first graphs and drawing force diagrams should help students qualitatively reason the change in the ball's path. If done quantitatively, assume the initial velocity of the ball is 90 mph; the ball's mass is .5 kg; it leaves the pitcher's hand at a height of 6 feet; and the distance from mound to plate is 60 ft-6 inches. Plot the position each tenth of a second.
Working through conditions 1 and 2 as a class may help clarify objectives for students.
After the activity has been completed, have the class or small groups discuss the following:
How does this model help to view the effects of smaller air resistance forces. How is the effect of gravity "controlled" in the model? How are the drag and Magnus forces modelled? How is this model similar to and different from the wind tunnel model in the video? How is this model similar to and different from a real thrown ball?
What were the effects of the "small" forces on the motion of the balloon? How would such forces affect the motion of an actual ball? If the effect of gravity was not "controlled", what would be the combined result of the forces on the ball?
Due to similar densities the water balloon should not rise or sink in the water. This is analogous to the "balanced" non-spinning ball in the wind tunnel. The Magnus forces are modelled by the small forces applied by the paper clips and the helium balloon. The major difference in this model arises from the motion of the water balloon through the water, as opposed to the stationary ball in the wind tunnel. The resistance to forward motion due to the water is not controlled as in the wind tunnel. Although qualitatively similar, the fluid resistance to such a large object is much greater than air resistance, thus the forward motion is affected more dramatically. The fluid resistance retards the effects of the upward and downward forces, allowing them to be viewed over a longer time.
Students can invite a local high school or college baseball or softball pitcher to demonstrate different pitches and explain how the proper spin is achieved.
While riding in a car, students can note the change in air resistance as the car picks up speed. A small increase can also be noted when the hand is "pulled" forward through the wind. This should give students real experience with how air resistance changes with speed, and thus how the different sides of a spinning ball receive different forces.
Students can hit golf balls at a driving range. When a ball is "sliced" or "hooked", it is given a side spin which alters the direction of the ball. If the ball were simply aimed at an angle but hit with no spin, it would travel in a straight line. A sliced ball curves dramatically as it travels. The concepts of a slice and a curveball are essentially the same, although the effects on a golf ball are more noticeable due to the longer time of flight.
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 Program #4 deals with the change from a pitcher's game to a hitter's game in the Twenties and Thirties caused by the regulation of the ball. Another segment deals with pitchers' attempts to subvert the rules by scuffing, spitting on, greasing, and otherwise altering the surface of the ball.
History/Social Studies: Student can study the influence of the labor movement and collective bargaining on the salaries of great hitters and pitchers in baseball.
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 analysis of motion at the end of the activity provides an excellent application of both graphing skills and work with quadratic equations.
1995-1996 National Teacher Training Institute / Austin
Master Teacher: Brendan Maxcy