About
Ball in Wedge
Ball in Wedge models the dynamics of a ball falling and bouncing elastically in a symmetric wedge. Although the ball's free-fall trajectory and bounce are simple and are discussed in introductory physics textbooks, the ball-wedge system exhibits a rich array of periodic and chaotic behavior and is of current research interest.
The ball has both chaotic and periodic trajectories but it is difficult to find initial conditions (x0, vx0, y0, vy0) that lead to periodicity. The "trick" is to examine the dynamics when the ball collides with the wall. At the time of collision, we record the ball's velocity component tangent to the wall vr and its velocity component perpendicular to the wall vn. A plot of vr vs. vn2 at the time of impact, known as a Poincaré section, provides a stroboscopic snapshot of the ball's state. The measured velocity components determine the ball's position because we can compute the ball's kinetic energy and use conservation of energy to compute the ball's height. The x-coordinate is computed using the y-coordinate and the wedge angle.
Run the Ball in Wedge model and observe the patterns. If the motion is periodic, the snapshot (Poincaré section) will show only a small number of points corresponding the number of collisions within the periodic orbit. If the trajectory is chaotic there the points on the plot will appear to be random. Initial conditions are selected by dragging the ball or by clicking within the Poincaré plot. Notice the symmetry of the patterns. Clicking at the center of an oval reveals the periodic trajectory.
Translations
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Credits
dieter.roess@t-online.de - WEH-Foundation; Fremont Teng; Loo Kang Wee; Francisco Esquembre
Executive Summary:
The "Ball in Wedge" JavaScript simulation applet, developed by Open Educational Resources / Open Source Physics @ Singapore, provides an interactive tool to visualize the complex dynamics arising from a simple physical system: a ball bouncing elastically within a symmetric wedge. Despite the elementary nature of the individual free-fall and bounce events, the system exhibits a rich array of both periodic and chaotic trajectories. The simulation allows users to explore these behaviors and introduces the concept of the Poincaré section as a method for analyzing the system's state at each collision with the wedge walls. This resource is intended for learning and teaching mathematics and physics, particularly concepts related to differential equations, dynamics, and chaos theory.
Main Themes and Important Ideas/Facts:
- Complex Dynamics from Simple Rules: The core theme is the emergence of complex behavior from a seemingly straightforward system. The source explicitly states: "Although the ball's free-fall trajectory and bounce are simple and are discussed in introductory physics textbooks, the ball-wedge system exhibits a rich array of periodic and chaotic behavior and is of current research interest." This highlights that even with well-understood individual components (gravity, elastic collisions), their interaction within the constrained geometry of the wedge leads to non-trivial outcomes.
- Periodic and Chaotic Trajectories: The simulation demonstrates that the ball's motion can be either periodic, where the trajectory repeats over time, or chaotic, where the trajectory appears random and is highly sensitive to initial conditions. The source notes: "The ball has both chaotic and periodic trajectories but it is difficult to find initial conditions (x0, vx0, y0, vy0) that lead to periodicity." This underscores the complexity in predicting the long-term behavior of the system.
- Poincaré Section for Analysis: A key analytical tool introduced by the simulation is the Poincaré section. The source explains: "At the time of collision, we record the ball's velocity component tangent to the wall vr and its velocity component perpendicular to the wall vn. A plot of vr vs. vn2 at the time of impact, known as a Poincaré section, provides a stroboscopic snapshot of the ball's state." This method allows for a simplified view of the system's evolution by focusing only on the state at discrete events (collisions), making patterns in the motion (periodic or chaotic) more discernible.
- Identifying Periodic and Chaotic Motion through Poincaré Sections: The appearance of the Poincaré section directly indicates the nature of the trajectory. "If the motion is periodic, the snapshot (Poincaré section) will show only a small number of points corresponding the number of collisions within the periodic orbit. If the trajectory is chaotic there the points on the plot will appear to be random." This provides a visual and qualitative way to distinguish between the two types of dynamical behavior.
- Role of Initial Conditions: The behavior of the ball-wedge system is highly dependent on the initial conditions (position and velocity of the ball). While not explicitly detailed in the instructions, the description implies that different starting points will lead to different trajectories, some periodic and many chaotic. The statement that finding periodic trajectories is "difficult" reinforces this sensitivity.
- Interactive Exploration and Learning: The applet is designed for interactive learning. Users can "Run the Ball in Wedge model and observe the patterns." They can also manipulate initial conditions by "dragging the ball or by clicking within the Poincaré plot." Furthermore, the "Adjustable Wedge" feature allows users to explore how the geometry of the system influences the dynamics, as it's noted that "the trails of the particle will differ based on the angle between the two wedges."
- Symmetry: The source points out the "symmetry of the patterns" observed in the simulation, although it doesn't elaborate on the nature or implications of this symmetry. This suggests a potentially important aspect of the system's behavior that users can explore.
- Technical Implementation: The simulation is a "JavaScript Simulation Applet HTML5," indicating its accessibility through web browsers without the need for additional plugins. The embed code provided allows for easy integration of the model into other web pages.
- Educational Context: The applet is listed under "Learning and Teaching Mathematics using Simulations – Plus 2000 Examples from Physics," clearly positioning it as an educational resource. The inclusion under "Differential equations" suggests a connection to mathematical modeling of physical systems. Sample learning goals and a section "For Teachers" further emphasize its pedagogical purpose.
- Credits and Licensing: The developers are credited, and the content is licensed under a Creative Commons Attribution-Share Alike 4.0 Singapore License, promoting open access and sharing for non-commercial purposes. Commercial use requires separate licensing.
Key Quotes:
- "Ball in Wedge models the dynamics of a ball falling and bouncing elastically in a symmetric wedge... the ball-wedge system exhibits a rich array of periodic and chaotic behavior and is of current research interest."
- "The ball has both chaotic and periodic trajectories but it is difficult to find initial conditions (x0, vx0, y0, vy0) that lead to periodicity."
- "A plot of vr vs. vn2 at the time of impact, known as a Poincaré section, provides a stroboscopic snapshot of the ball's state."
- "If the motion is periodic, the snapshot (Poincaré section) will show only a small number of points... If the trajectory is chaotic there the points on the plot will appear to be random."
- "Initial conditions are selected by dragging the ball or by clicking within the Poincaré plot."
- "Notices that the trails of the particle will differ based on the angle between the two wedges."
Conclusion:
The "Ball in Wedge" JavaScript simulation applet is a valuable open educational resource for exploring fundamental concepts in dynamics and chaos theory. Its interactive nature, coupled with the visualization of Poincaré sections, provides a powerful tool for students and educators to understand how simple physical rules can lead to complex and unpredictable behaviors. The applet encourages experimentation with initial conditions and system parameters (wedge angle) to discover the transitions between periodic and chaotic regimes. The resource's accessibility through standard web browsers further enhances its utility in educational settings.
Ball in Wedge Simulation Study Guide
Key Concepts
- Elastic Collision: A collision in which there is no net loss in kinetic energy in the system as a result of the collision. Both momentum and kinetic energy are conserved.
- Free-fall Trajectory: The path an object follows under the influence of gravity alone, without any other forces acting upon it.
- Periodic Behavior: Motion that repeats itself in regular intervals of time. In the context of the ball in the wedge, this would mean the ball follows the same path and bounces in the same way repeatedly.
- Chaotic Behavior: Motion that is highly sensitive to initial conditions. Even very small changes in the starting position or velocity of the ball can lead to drastically different long-term trajectories. This behavior often appears random and unpredictable.
- Initial Conditions (x0, vx0, y0, vy0): The starting position (x0, y0) and velocity components (vx0, vy0) of the ball at the beginning of the simulation. These conditions determine the subsequent motion of the ball.
- Collision Dynamics: The physics governing the interaction between the ball and the wedge walls during a bounce, particularly the change in velocity components.
- Velocity Component Tangent to the Wall (vr): The component of the ball's velocity that is parallel to the surface of the wedge at the point of impact.
- Velocity Component Perpendicular to the Wall (vn): The component of the ball's velocity that is perpendicular to the surface of the wedge at the point of impact. This component reverses direction (and may change magnitude depending on elasticity) during an elastic collision.
- Poincaré Section: A stroboscopic snapshot of the system's state at discrete times, typically at the moment of collision. In this simulation, it is a plot of the tangential velocity (vr) versus the square of the perpendicular velocity (vn²) at each impact with the wedge.
- Stroboscopic Snapshot: A view of a moving system sampled at regular intervals, like using a flashing strobe light to observe motion. The Poincaré section provides such a snapshot of the ball's velocity at each collision.
- Kinetic Energy: The energy of motion, given by 1/2 * mass * velocity².
- Conservation of Energy: The principle that the total energy of an isolated system remains constant over time. In the ball-wedge system, the sum of the ball's kinetic and potential energy (due to height) remains constant (assuming elastic collisions and no energy loss).
- Wedge Angle: The angle formed by the two surfaces of the symmetric wedge. This angle influences the possible trajectories and the dynamics of the collisions.
- Simulation Applet: A small, self-contained program (in this case, written in JavaScript and HTML5) that models a physical system and allows users to interact with it.
Short Answer Quiz
- What fundamental physics principles govern the motion of the ball between bounces in the Ball in Wedge simulation?
- Explain the difference between periodic and chaotic behavior in the context of the Ball in Wedge simulation.
- What specific information about the ball's motion is recorded at the time of collision with the wedge walls? Why is this information important?
- Describe what a Poincaré section is in the context of this simulation and what it reveals about the ball's trajectory.
- How can a user interact with the Ball in Wedge simulation to change the initial conditions of the ball?
- According to the text, what visual characteristic of the Poincaré section indicates periodic motion of the ball? What indicates chaotic motion?
- Explain how the ball's kinetic energy and the principle of conservation of energy are used to determine the ball's height at the time of impact.
- How is the x-coordinate of the ball determined at the time of collision, based on the information provided in the text?
- What happens to the trails of the particle in the simulation when the angle between the two wedges is adjusted?
- What do the "Play/Pause," "Step," and "Reset" buttons in the simulation allow the user to do?
Short Answer Quiz Answer Key
- Between bounces, the ball's motion is governed by gravity, resulting in a free-fall trajectory described by kinematic equations. The bounces themselves are governed by the principles of elastic collisions, where both kinetic energy and momentum are conserved.
- Periodic behavior means the ball's motion repeats in a predictable cycle over time, following the same path and bouncing pattern. Chaotic behavior, on the other hand, is highly sensitive to initial conditions and appears random and unpredictable, with trajectories that do not repeat regularly.
- At the time of collision, the simulation records the ball's velocity component tangent to the wall (vr) and its velocity component perpendicular to the wall (vn). This information is crucial for analyzing the dynamics of the collision and constructing the Poincaré section.
- A Poincaré section is a plot of the tangential velocity (vr) versus the square of the perpendicular velocity (vn²) at the moment of each impact. It provides a stroboscopic snapshot that helps visualize the underlying structure of the ball's motion, distinguishing between periodic and chaotic regimes.
- Users can change the initial conditions of the ball by directly dragging the ball to a new starting position within the simulation window or by clicking within the Poincaré plot, which corresponds to selecting specific velocity components at impact.
- In the Poincaré section, periodic motion is indicated by a small number of discrete points, corresponding to the limited number of unique collision states within a repeating orbit. Chaotic motion is indicated by a seemingly random scattering of points across the plot.
- The ball's kinetic energy at the time of impact can be calculated from its velocity. Due to the conservation of energy, this kinetic energy can be related to the ball's potential energy (mgh) at its maximum height, thus allowing the computation of the height.
- The x-coordinate of the ball at the time of collision is computed using the determined y-coordinate (height) and the angle of the symmetric wedge, due to the geometric constraints of the wedge shape.
- Adjusting the angle between the two wedges will cause the trails of the particle to differ because the angle affects the direction of the normal force during the collision, thus changing the subsequent trajectory of the ball.
- The "Play/Pause" button starts and stops the simulation, the "Step" button advances the simulation by a single time increment, and the "Reset" button returns the simulation to its initial state.
Essay Format Questions
- Discuss the significance of the Poincaré section as a tool for analyzing the dynamics of the Ball in Wedge system. How does it help distinguish between periodic and chaotic behavior, and what information does it provide about the system's state?
- Explain how the interplay between simple gravitational free-fall and elastic collisions in the Ball in Wedge system can lead to complex behaviors such as periodicity and chaos. What factors contribute to the emergence of these different types of motion?
- Consider the role of initial conditions in determining the trajectory of the ball in the wedge. How sensitive is the system to changes in these conditions, and how does this sensitivity manifest in both periodic and chaotic regimes?
- Describe the different ways a user can interact with the Ball in Wedge JavaScript Simulation Applet HTML5. How do these interactive features enhance the learning and understanding of the physical concepts being modeled?
- Based on the provided text, discuss the potential educational value of the Ball in Wedge simulation for teaching concepts in physics and mathematics. What specific learning goals might this simulation help students achieve?
Glossary of Key Terms
- Differential Equations: Mathematical equations that relate a function with its derivatives. They are often used to model systems that change over time, such as the motion of the ball in the wedge.
- Dynamics: The branch of mechanics concerned with the motion of objects and the forces that cause that motion. The Ball in Wedge model explores the dynamics of a bouncing ball within a constrained environment.
- Symmetric Wedge: A wedge shape where both sides form the same angle with a central axis or plane. In this simulation, the wedge is described as symmetric, implying equal angles for both inclines.
- Trajectory: The path followed by a moving object, such as the ball as it moves and bounces within the wedge.
- Kinematic Equations: A set of equations that describe the motion of an object without considering the forces that cause the motion. These equations relate displacement, velocity, acceleration, and time during the free-fall portions of the ball's trajectory.
Sample Learning Goals
[text]
For Teachers
Ball in Wedge JavaScript Simulation Applet HTML5
The ball has both chaotic and periodic trajectories but it is difficult to find initial conditions (x0, vx0, y0, vy0) that lead to periodicity. The "trick" is to examine the dynamics when the ball collides with the wall. At the time of collision, we record the ball's velocity component tangent to the wall vr and its velocity component perpendicular to the wall vn. A plot of vr vs. vn2 at the time of impact, known as a Poincaré section, provides a stroboscopic snapshot of the ball's state. The measured velocity components determine the ball's position because we can compute the ball's kinetic energy and use conservation of energy to compute the ball's height. The x-coordinate is computed using the y-coordinate and the wedge angle.
Run the Ball in Wedge model and observe the patterns. If the motion is periodic, the snapshot (Poincaré section) will show only a small number of points corresponding the number of collisions within the periodic orbit. If the trajectory is chaotic there the points on the plot will appear to be random. Initial conditions are selected by dragging the ball or by clicking within the Poincaré plot. Notice the symmetry of the patterns. Clicking at the center of an oval reveals the periodic trajectory.
Instructions
High Speed Check Box
Checking this will set the speed of the simulation to be fast or slow.
Adjustable Wedge
Toggling Full Screen
Play/Pause, Step and Reset Buttons
Research
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Other Resources
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Frequently Asked Questions: Ball in Wedge Simulation
What is the "Ball in Wedge" simulation about?
The Ball in Wedge simulation models the physical system of a ball falling and bouncing elastically within a symmetric wedge. While the individual free-fall and bounce mechanics are straightforward concepts typically covered in introductory physics, the combined system can exhibit a wide variety of both periodic and chaotic behaviors, making it a subject of ongoing research interest.
What makes the behavior of the ball in the wedge complex?
Despite the simplicity of the individual components (gravity and elastic collisions), the repeated interactions between the ball and the wedge walls, governed by initial conditions, lead to complex dynamics. These dynamics can manifest as predictable, repeating patterns (periodic motion) or as seemingly random and unpredictable motion (chaotic behavior).
How can I observe the different types of motion in the simulation?
By running the Ball in Wedge model, you can observe the patterns traced by the ball over time. Periodic motion will show the ball following a repeating path. Chaotic motion, on the other hand, will appear irregular and non-repeating. You can adjust the initial conditions by dragging the ball or clicking within the Poincaré plot to explore different types of trajectories.
What is a Poincaré section and what does it reveal about the ball's motion?
A Poincaré section is a specific way of analyzing the system's state at discrete moments in time – specifically, at the moment of each collision with the wedge wall. It is a plot of the ball's velocity component tangent to the wall (vr) versus the square of its velocity component perpendicular to the wall (v2n) at the instant of impact. This stroboscopic snapshot helps to identify whether the motion is periodic (showing a limited number of points on the plot) or chaotic (showing a seemingly random scattering of points).
How are the ball's position and velocity tracked and related?
At the time of each collision, the simulation records the ball's tangential velocity (vr) and perpendicular velocity (vn). These velocities are crucial because they allow for the calculation of the ball's kinetic energy at the moment of impact. Using the principle of conservation of energy, the ball's height can then be determined. The x-coordinate of the ball is subsequently calculated based on its y-coordinate (height) and the angle of the wedge.
How can I control the simulation and explore different scenarios?
The simulation provides several controls: a "High Speed" checkbox to adjust the simulation speed, draggable red boxes to modify the angle of the wedge, a toggle for full-screen mode (activated by double-clicking within the panel), and standard "Play/Pause," "Step," and "Reset" buttons to manage the simulation's progression. You can also directly influence the initial conditions by clicking and dragging the ball within the simulation window or by clicking within the Poincaré plot.
What is the significance of the symmetry observed in the Poincaré sections?
The patterns displayed in the Poincaré sections often exhibit symmetry. This symmetry reflects underlying symmetries in the physical system itself, such as the symmetric geometry of the wedge and the nature of the elastic collisions. Observing these symmetries can provide insights into the fundamental dynamics governing the ball's motion.
Who developed this simulation and where can I find more resources?
This Ball in Wedge JavaScript Simulation Applet HTML5 was developed by contributors including the WEH-Foundation, Fremont Teng, Loo Kang Wee, and Francisco Esquembre, as part of the Open Educational Resources / Open Source Physics @ Singapore project. The platform hosts a wide range of other interactive physics and mathematics simulations, which can be explored through the provided links and tags. For commercial use of the EasyJavaScriptSimulations Library used in this applet, specific licensing information and contact details are provided.
- Details
- Written by Fremont
- Parent Category: 5 Calculus
- Category: 5.5 Differential equations
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