Learning Scenario - Bouncing Ball Model (AgentSheets)

Basic Model:

Description

The bouncing ball model is a basic model that helps students develop an understanding of complex physics and math concepts, such as inelastic collisions and reflections. The user places a number of balls into a box and then they bounce off of the walls with the program-set initial velocity. The pattern and path of the balls may be surveyed in order to deduce patterns and trends related to transformations, translations, and collisions. The main equation investigated here is the angle of incidence is equal to the angle of reflection.

Background Information

Balls in this model do not interact with each other at all. They can move through each other and do not have any effect on the path of the other. The wall, on the other hand, bounces the balls at a precise angle. Since the angle of depression is always going to be a 45 degree angle, the angle of reflection will also be 45 degrees. These movement patterns are similar to the mathematical concepts of reflection and transformations; when an equation is reflected over a line, each point will be equidistant from the line as its counterpart on the other side. The red bouncing balls in this model can be seen as points in an equation and the wall as the reflection line. If the model is tied to chemistry, the balls can also be seen as ideal gas particles that do not have any interactions with each other and have perfectly elastic collisions with the barriers of the container.

Science/Math

The fundamental principle behind this model is HAVE = HAD + CHANGE. Essentially, the following changes will contribute to an understanding of the topic and the process between the HAVE and the HAD:

1. The balls preset by the user begin moving with an initial velocity at a 45 degree angle
2. The walls transfer 100 percent of the energy back to the balls and reflect them at a 45 degree angle

The CHANGE in this model is not very user-intensive, but the processes allow for a better understanding of concepts in physics, chemistry, and mathematics.

Teaching Strategies

An effective way of introducing this model is to ask students to think about the concepts introduced in the model and find what preconceptions they have about the subject. The questions might be best complemented by an activity with real balls and a box. By throwing the ball around in an actual box students may begin to develop an understanding of the actual physics involved in the model.

1. What do you think will happen when the ball hits the side of the box?
2. Where do you think it will go after that?
3. Do you think it will slow down? If so, when?
4. Imagine this was a bottle with gas particles in it. Would they behave differently or similarly?

Implementation:

How to use the model

This relatively in-depth model only has one parameter that may be changed, but it is important to study the changes resulting from it. Any number of balls may be placed into the box and sent to bounce around. To insert balls, click the pencil icon and draw them inside the box. Next, run the simulation by clicking the green triangle in the bottom left hand corner of AgentSheets. Instantly, the balls will begin to bounce around the container until they are stopped. To reset the simulation, click the "Reset" button in the bottom right hand corner of the program, and to slow it down, drag the slider at the bottom. For more information about AgentSheets reference the AgentSheets tutorial at:

http://shodor.org/tutorials/agentSheets/Introduction

Learning Objectives

1. Understand the mathematical concept of reflections and translations
2. Understand the physics concept of inelastic and elastic collisions

Objective 1

In mathematics, the reflection of a function will place a similar function on the other side of a line with each reflected point equidistant from the line as its counterpart. This model will allow students to reflect the points off of the wall and see the resulting similarities. Ask the following questions:

1. Start with a point in the lower left hand section of the worksheet and turn the speed to moderately slow. Watch as the ball bounces off the left wall. What is the trajectory when it bounces? What is the relationship between the angle of depression and the angle of reflection?
2. Make an axes out of balls and draw a function on top of it (see picture). How does the function seem to change when the graph bounces off of the walls? What line does the function seem to be reflected across?

3. If the balls represent a function, how would you move the function as the model does? What variables or constants would you manipulate?
4. Could the balls ever change their angle of reflection? Why or why not?

Objective 2

In the real world, when objects collide with each other, they will squish, bend, or otherwise move. Ideal situations, though, as in this model, do not calculate in any secondary transfer of energy. 100 percent of the kinetic energy that hits the wall is transferred back to the ball. This is known as a perfectly elastic collision. With inelastic collisions, the energy is converted from kinetic energy to heat, friction, deformation, etc. Students will be able to understand these concepts by comparing the simulation with the real world. Ask the following questions:

1. Does the simulation seem to be slowing down? What happens to the velocity of the balls when they hit the walls? Why?
2. If you were to roll a ball into a wall, what would eventually happen to it? Why? What is missing from this model?
3. Do the balls appear to change shape when they hit the wall? Does this happen in real life? Explain.
4. What can be said about the kinetic energy in this system? Is it transferred at all between objects or into other types of energy?
5. What could you change about this model to make it more resemble the real-world situations it represents?

Extensions and Related Models:

1. Extend the idea of elastic and inelastic collisions to the ideal gas laws
2. Explore transformations, translations, and reflections in relation to music

Extension 1

With an elastic collision, the kinetic energy is completely conserved in the system. The Ideal Gas Laws require both perfectly elastic collisions and particles that do not interact with each other for modeling purposes. Even though in real life gas particles will not always behave in this manner, they do emulate similar behavior in ideal situations. Have students tie the idea of the Ideal Gas Laws to the model. Ask the following questions:

1. If the box in this model was emulating a container filled with gas, what would the particles represent? Do the balls show similar characteristics?
2. How do the collisions of the balls against the walls simulate particles against the walls of a container? Do ideal gasses have elastic or inelastic collisions? Explain.
3. Do the particles of ideal gasses interact with each other? How well does the Bouncing Ball Model represent this?

Extension 2

Some students who use this model may play an instrument or have some connection with reading and playing music. This extension ties the two together and analyzes the geometric connections to music. Geometric translations, reflections, and other operations can be seen in music through key changes, refrains, and retrogrades. Have students read over the following article and connect it to the concepts learned with the Bouncing Ball Model. Ask the following questions:

1. What examples of translations do you see in music? What does it sound like? How are these translations related to the ones learned with the Bouncing Ball Model?
2. What type of reflection would turning music upside down be? What would be the line of reflection?
3. Can you think of songs you know that demonstrate any of the concepts discussed in the article? How do they demonstrate geometric operations?

Supplemental Material: Listening to Geometry by Brett D. Cooper and Rita Barger - http://alexbradley02.wikispaces.com/file/view/Listening+to+Geometry.pdf

Related Models

Ideal Gas Model

http://www.shodor.org/talks/ncsi/vensim/

The Ideal Gas Model starts with two connected flasks and allows the user to study the diffusion of gas particles from one container into another. The system is guided by the ideal gas laws and allows the gas particles to follow them with perfectly elastic collisions against the side of the canister. Ratios of the number of molecules to the volume are recorded as well. These may be studied for a better understanding of the gas laws and their equations. This model is a good connection for both the concepts of elastic collisions learned in the Bouncing Ball model and Extension 1 above.

Simple Function Model

http://www.shodor.org/talks/ncsi/agentsheets/

As the name suggests, the Simple Function Model starts with one equation and several constants that may be changed. The graph of the equation is displayed next to the model and shows the user what is or her manipulations of the variables accomplishes. This model is a good complement and next step from the Bouncing Ball Model. In the Bouncing Ball model, students simply studied the visual aspects of reflections and translations, but the Simple Function model will allow them to see how constants in an equation dictate the transformational behavior of a function.

Multi-Function Data Flyer

http://www.shodor.org/interactivate/activities/MultiFunctionDataFly/

The Simple Function Model provides a simple introduction to functions, but if a more advanced model is appropriate, the Multi-Function Data Flyer has several other tools available. While it still has the basic function input and constant manipulation, it also has the ability to trace functions so as to see the full extent that one constant has on the equation. Data plotting and deviations may also be useful for a statistical extension to a subject. All these tools will allow for a better understanding of transformations and expand the student's knowledge of the concepts introduced in the Bouncing Ball model.