Translations

Code	Language		Translator	Run

Credits

This email address is being protected from spambots. You need JavaScript enabled to view it.; Francisco Esquembre; Felix J. Garcia Clemente

This is an interactive HTML5 JavaScript applet designed to simulate the "pi-collision problem," a physics scenario where the number of elastic collisions between two blocks of differing masses is surprisingly related to the digits of π. This open educational resource from Open Source Physics @ Singapore includes an embeddable simulation, learning goals for teachers, and links to related research and videos, explaining the connection between the collision count and the mathematical constant. The page also serves as a repository for numerous other physics and math simulations created using JavaScript and Easy JavaScript Simulation, showcasing a wide range of interactive learning tools.

Briefing Document: The Pi-Collision Problem

This briefing document reviews the main themes and important ideas presented in two sources concerning the "pi-collision problem," a physics puzzle where the number of elastic collisions between two blocks and a wall surprisingly relates to the digits of pi.

Source 1: Excerpts from "There's more to those colliding blocks that compute pi" (YouTube video transcript excerpt)

This source, a transcript from a science communicator's video, revisits the well-known puzzle of two colliding blocks on a frictionless plane, one initially stationary and smaller (mass \(m_2\)), the other larger (mass \(m_1\)) moving towards it. The core question is the total number of collisions (including those with a wall to the left of the smaller block) until both blocks are moving to the right and the smaller block is slower.

Main Themes and Important Ideas:

• The Surprising Connection to Pi: The central and most popular aspect of the puzzle is that as the mass ratio (\(m_1/m_2\)) increases by powers of 100, the total number of collisions yields the digits of pi.
• "In this case, with a mass ratio of 10,000 to 1, that little block ends up getting really crammed up against the wall, and almost all of the collisions happen in this rapid burst right in the middle. In this case, it takes over 300 in total to turn that big block around, and after another nice, long pause, as if the universe has a strong taste for drama, we see that the exact answer rolls in at 314, a number with eerily familiar digits."
• "Like here for example, with a mass ratio of a million to one, that burst contains almost 3,000 collisions, and the final count comes in at 3,141."
• Idealized Classical Physics Puzzle: The video emphasizes that the connection to pi arises under highly idealized conditions, ignoring real-world complexities:
• "I do want to highlight a couple unrealistic things about this puzzle, the kind of nuances you can't fit into a short. ... So in order to get the delightful result here, we have to make a number of idealizing assumptions. One assumption is that no energy is lost in the collisions. Physicists would call those perfectly elastic collisions."
• Relativistic effects at very high mass ratios and the physicality of collisions (sound) are also ignored for the pure puzzle.
• Connection to Quantum Computing (Grover's Algorithm): A novel point introduced in this revisited video is a "secret" connection between this classical mechanics puzzle and Grover's Algorithm for search in quantum computing. This connection will be explored in a subsequent video.
• "this pure puzzle, unrealistic though it might be, is secretly connected to quantum computing. And believe it or not, studying this I think puts you in a good position to understand what quantum computing even is."
• The reason for mass ratios increasing by powers of 100, rather than 10, is "intimately connected with the fact that for quantum computers, search algorithms can be faster than they are for classical computers."
• Problem-Solving Principles: The puzzle serves as a lesson in general problem-solving:
• "What makes this puzzle great, aside from the surprising pi, of course, is how well poised it is to be a general problem solving lesson."
• Two principles highlighted are: listing relevant equations/theorems (conservation of energy and momentum) and drawing pictures/visualizing the problem.
• State Space and Geometric Interpretation: The key to understanding the connection to pi lies in analyzing the problem in an abstract velocity state space where the velocities of the two blocks are the coordinates.
• "Even though we're just getting warmed up, this really is the key step in our whole problem solving process... when you package all those numbers together as a single point in a higher dimensional space, studying how that point moves through the space has a way of clarifying the whole problem."
• Conservation of energy in this state space is represented by an ellipse, which can be transformed into a circle by rescaling the coordinates to represent the square root of mass times velocity.
• Transformation to a Circle Puzzle: The physics problem is translated into a purely geometric puzzle on a circle: starting at the leftmost point and repeatedly moving down-right along a line with a specific slope (related to the mass ratio), then straight up to the circle, until reaching an "end zone." The number of "zig-zag" moves corresponds to the number of collisions.
• "Our new puzzle is this. Imagine I said we have a circle. You start at the left most point of that circle and your first move is to travel down and to the right along a line with some specific slope... The puzzle is, how many lines do you draw? How many times do you zig and zag throughout this process?"
• The Role of the Inscribed Angle Theorem and Small Angle Approximation: The number of lines drawn in the circle puzzle relates to how many equal-sized arcs can fit along the semicircle. The angle of these arcs is related to the slope of the line, which in turn depends on the mass ratio.
• The inscribed angle theorem helps prove that all the arcs created by the intersections are of equal size.
• The tangent of a small angle (theta) is approximately equal to the angle itself (in radians). This "small angle approximation" connects the arctangent of the square root of the inverse mass ratio (which determines the angle) to the power of 10 in the mass ratio, ultimately leading to the digits of pi.
• "The key idea is that for the purpose of our question, where we're saying what's the biggest whole number that you can multiply by this little angle that keeps it from surpassing pi, these numbers are so close that they might as well be the same. You get the same whole number answer."
• The Unsolved Problem: The exact mathematical proof that the number of collisions perfectly matches the digits of pi for arbitrarily large mass ratios (powers of 100) is technically unsolved due to the uncertainty in the digit sequence of pi (specifically the possibility of an infinite sequence of nines that could cause an off-by-one error).
• "So very technically speaking, this fact that colliding blocks with a mass ratio that's a power of one hundred can compute pi for you is an unsolved problem."
• Generalizability to Other Bases: The principle is not specific to base 10. If working in base two, mass ratios with a power of four would relate to the binary digits of pi.
• Justification for Idealization: Idealizing the problem, despite its unrealistic nature, is valuable for two reasons: it's a helpful first step in solving complex problems, and it can expose hidden connections between seemingly disparate areas of mathematics and physics.
• "The deeper reason for abstracting away from the messiness of the real world is that purity can expose hidden connections."

Source 2: Excerpts from "pi-collision problem simulator html5 JavaScript applet - Open Educational Resources / Open Source Physics @ Singapore" (Webpage)

This source provides a brief overview of the pi-collision problem and offers a JavaScript simulator.

Main Themes and Important Ideas:

• Simulation Tool: The primary offering is an embeddable HTML5 JavaScript applet that allows users to simulate the pi-collision problem.
• Includes an iframe embed code for easy integration into webpages.
• Reinforcement of the Pi Connection: The text explicitly states the link between the number of collisions and pi.
• "The number of collisions in this setup, often referred to as the 'pi-collision problem' or 'digit dynamics,' tends to be around 3140. This intriguing result is related to the mathematical constant π (pi)." (Note: This example with 3140 seems to imply a mass ratio of \(100^3 = 1,000,000\))
• Specific Examples and Collision Counts: The page provides concrete examples of mass ratios and their corresponding collision counts:
• m1 = m2 = 1, collisionCounter = 3
• m1 = m2 = 100, collisionCounter = 31
• m1 = m2 = 10000, collisionCounter = 314
• Relationship to Mass Ratio (1:100^n): The source clearly states the general rule for the connection to pi's digits based on the mass ratio.
• "For two blocks with masses in the ratio 1:100n, the number of collisions will be approximately equal to the first n+1 digits of π. In this case, with n = 3 (since 10000 = 1002), the number of collisions is close to the first 3 digits of π, which is 314." (Correction: if \(n=2\) for \(100^2 = 10000\), then \(n+1 = 3\) digits. If the ratio is \(1:100^n\), and \(m_1\) is the larger mass, then for \(m_1/m_2 = 100^n\), the number of collisions relates to the first \(n+1\) digits.)
• Explanation via Iterative Calculations: The connection to pi is attributed to the nature of the collisions and velocities, involving iterative calculations that "mimic the series expansion of π."
• "This remarkable connection arises from the way the collisions and velocities behave, involving iterative calculations that mimic the series expansion of π."
• Computational Intensity: The page acknowledges the computational challenges for very high mass ratios.
• The question "How to get to 1000000 kg? and collisionCounter = 3141 ?" and the subsequent suggestion to slow down the velocity indicate the practical limitations of simulating a large number of collisions.
• Links to Video Resources: The page provides links to relevant YouTube videos, including one of the sources reviewed (likely the original video).

Overall Summary:

Both sources highlight the fascinating pi-collision problem. The video transcript offers a detailed explanation of the underlying physics and the surprising mathematical connection to pi, delving into the idealized nature of the problem, the geometric interpretation in state space, and even hinting at a connection to quantum computing. It also addresses the technically "unsolved" nature of a rigorous proof for arbitrarily large mass ratios. The webpage provides a practical, interactive simulation of the problem and reinforces the empirical observation of the collision count approaching the digits of pi for specific mass ratios, attributing this to iterative calculations. Together, they provide both a theoretical understanding and a hands-on way to explore this intriguing physics puzzle.

Study Guide: The Pi-Colliding Blocks Problem

Overview

This study guide is designed to help you review your understanding of the fascinating connection between colliding blocks and the mathematical constant pi, as well as its surprising link to quantum computing. It covers the setup of the problem, the mathematical and physical principles involved, the geometric interpretation, and the implications of this phenomenon.

Key Concepts to Understand

• The Setup: Two blocks of different masses on a frictionless plane, one initially stationary and the other moving towards it. A wall is located to the left of the smaller block.
• Collisions: Perfectly elastic collisions where no energy is lost.
• Conservation Laws: Conservation of energy (kinetic energy before and after collisions remains constant) and conservation of momentum (total momentum before and after collisions remains constant, except when the small block hits the wall).
• Mass Ratio: The ratio between the mass of the larger block (m1) and the smaller block (m2) is crucial. The number of collisions relates to pi when this ratio is a power of 100.
• State Space: Representing the velocities of the two blocks as a point in a 2D space (v1, v2) or a rescaled space (sqrt(m1)v1, sqrt(m2)v2).
• Geometric Interpretation: The conservation of energy in the rescaled state space corresponds to a circle. Collisions between the blocks and the wall can be visualized as the movement of a point along this circle, constrained by lines representing conservation of momentum.
• The End Zone: The region in the state space where both blocks are moving to the right and the smaller block is slower than the larger block, signifying the end of the collisions.
• Inscribed Angle Theorem: A geometric theorem that relates the angle subtended by an arc at the circumference of a circle to the angle subtended by the same arc at the center.
• Small Angle Approximation: For small angles (theta), the tangent of the angle (tan θ) is approximately equal to the angle itself (θ), measured in radians.
• Connection to Pi: The number of collisions is related to how many small arcs of a certain angle (related to the arctangent of the square root of the inverse mass ratio) can fit along the circumference of the state space circle. This number approaches pi as the mass ratio increases by powers of 100.
• Quantum Computing Connection: The problem has a surprising analogy to Grover's Algorithm for Search in quantum computing.
• Unsolved Problem Aspect: The exact mathematical proof that the number of collisions perfectly mirrors the digits of pi for infinitely large mass ratios that are powers of 100 is technically an unsolved problem due to the nature of pi's digits.
• Idealizations: The puzzle relies on several unrealistic assumptions, such as perfectly elastic collisions and a frictionless plane, to arrive at the clean mathematical result.

Quiz

Answer the following questions in 2-3 sentences each.

1. Describe the initial setup of the two blocks in the pi-colliding blocks problem.
2. What are the two fundamental laws of classical physics that govern the collisions between the blocks?
3. How does the concept of a "state space" help in analyzing the colliding blocks problem? What do the axes of this space typically represent?
4. In the rescaled state space, what geometric shape does the conservation of energy correspond to? Why is this change of variables useful?
5. Explain how the collision of the smaller block with the wall is represented in the state space diagram.
6. What is the "end zone" in the state space, and what physical condition of the blocks does it represent?
7. How does the slope of the line segment representing the collision between the blocks relate to the masses of the blocks?
8. Explain the role of the inscribed angle theorem in understanding why the arcs in the geometric interpretation are of equal size.
9. What is the small angle approximation, and why is it relevant to connecting the number of collisions to pi?
10. Briefly describe the surprising connection mentioned in the source material between the pi-colliding blocks problem and quantum computing.

Quiz Answer Key

1. The setup involves two blocks on a frictionless plane. Initially, the smaller block is stationary on the left, and the larger block is moving towards it from the right. There is also a wall to the left of the smaller block.
2. The two fundamental laws are the conservation of energy and the conservation of momentum. Conservation of energy states that the total kinetic energy of the system remains constant during collisions, while conservation of momentum states that the total momentum remains constant (excluding the interaction with the wall).
3. A state space is a coordinate plane where the axes represent the velocities (or rescaled velocities) of the two blocks. Tracking the movement of a point in this space as the blocks collide helps visualize the system's evolution and reveal underlying patterns.
4. In the rescaled state space (with axes sqrt(m1)v1 and sqrt(m2)v2), the conservation of energy corresponds to a circle. This change of variables introduces more symmetry into the problem, making the geometric interpretation simpler.
5. When the smaller block collides with the wall, its velocity reverses direction. In the state space diagram, this corresponds to a reflection of the point across the vertical axis (or the axis representing the larger block's rescaled velocity).
6. The end zone is a region in the state space where the x and y coordinates are both positive (both blocks moving right) and the y-coordinate is smaller than a certain proportion of the x-coordinate (small block moving slower). When the state space point enters this zone, the experiment is over.
7. The slope of the line segment representing the collision between the blocks in the rescaled state space is the negative square root of the mass ratio of the smaller block to the larger block (-sqrt(m2/m1)). This slope determines the angle of each "zig-zag" movement.
8. The inscribed angle theorem shows that angles subtended by the same arc on the circumference are equal. This implies that each collision sequence (block-block then block-wall) corresponds to traversing equal-sized arcs along the state space circle.
9. The small angle approximation states that for small angles, tan(θ) ≈ θ (in radians). This is relevant because the angle related to the slope of the collision line is such that its tangent is proportional to a small power of 10 (for mass ratios that are powers of 100), making the angle itself approximately that small power of 10.
10. The pi-colliding blocks problem, particularly its geometric interpretation in state space, has a surprising analogy to Grover's Algorithm, a quantum algorithm for searching unsorted databases. Both involve a sequence of reflections in a multi-dimensional space that leads to a desired outcome after a number of steps related to the geometry of the space.

Essay Format Questions

Consider the following questions for essay format answers. You do not need to provide answers here, but think critically about how you would structure your response using the information from the source material.

1. Discuss the role of idealizations in the pi-colliding blocks problem. While these simplifications allow for the elegant connection to pi to emerge, what are some of the practical limitations and real-world factors that would affect the outcome of a physical experiment?
2. Explain the transformation from the physical collision problem to the geometric puzzle involving a circle and lines in state space. What are the key correspondences between the physical quantities (masses, velocities, collisions) and the elements of the geometric representation (circle, slope, intersections)?
3. The source material highlights a connection between this classical physics puzzle and quantum computing, specifically Grover's Algorithm. Based on your understanding of the colliding blocks, speculate on potential parallels or analogous concepts that might exist between these two seemingly disparate fields.
4. The fact that the connection between the colliding blocks and pi is technically an "unsolved problem" is discussed in the source. Elaborate on why this is the case, focusing on the properties of pi and the limitations of current mathematical proofs regarding its digits.
5. The author suggests that studying seemingly unrealistic problems like the colliding blocks can still be valuable for problem-solving skills and revealing hidden connections. Discuss the broader implications of this idea, providing examples from the source material of the benefits of abstraction and the pursuit of purity in mathematical and scientific inquiry.

Glossary of Key Terms

• Elastic Collision: A collision in which there is no net loss in kinetic energy in the system as a result of the collision.
• Frictionless Plane: A theoretical surface with no resistance to motion.
• Conservation of Energy: A fundamental principle in physics stating that the total energy of an isolated system remains constant over time. In this context, the total kinetic energy of the two blocks before and after each elastic collision is the same.
• Conservation of Momentum: A fundamental principle in physics stating that the total momentum of an isolated system remains constant if no external forces act upon it. In this context, the total momentum of the two blocks before and after each collision (excluding the wall) is the same.
• Mass Ratio: The ratio of one mass to another. In this problem, the ratio between the larger block's mass and the smaller block's mass is crucial.
• State Space: An abstract space in which the state of a physical system is represented by a point. For the colliding blocks, the state can be defined by the velocities of the two blocks.
• Trajectory: The path followed by a point moving in space as time progresses. In the state space, the trajectory represents the changing velocities of the blocks during the collisions.
• Inscribed Angle Theorem: In a circle, an angle subtended by a chord at the circumference is half the angle subtended by the same chord at the center.
• Radian: A unit of angular measurement equal to the angle subtended at the center of a circle by an arc equal in length to the radius of the circle (approximately 57.3 degrees).
• Small Angle Approximation: The approximation that for small angles θ (in radians), sin(θ) ≈ θ, tan(θ) ≈ θ, and cos(θ) ≈ 1.
• Arctangent (arctan): The inverse trigonometric function of the tangent. arctan(y) gives the angle whose tangent is y.
• Grover's Algorithm: A quantum algorithm for searching an unsorted database with N entries in O(√N) time, providing a quadratic speedup over classical algorithms.
• Taylor Series (or Taylor Expansion): A representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The Taylor series for tan(θ) includes a θ³ term, indicating the error in the small angle approximation.

Sample Learning Goals

computational skills such as to add collisionCounter to an existing simulation to realise the computational goal of checking the number of collisions in this "pi-collision problem"

For Teachers

The number of collisions in this setup, often referred to as the "pi-collision problem" or "digit dynamics," tends to be around 3140. This intriguing result is related to the mathematical constant π (pi).

m1 = m2 = 1, collisionCounter = 3 link

 

m1 = m2 = 100, collisionCounter = 31 link

 

 

m1 = m2 = 10000, collisionCounter = 314 link

 

"pi-collision problem"

"digit dynamics," tends to be around 314. This intriguing result is related to the mathematical constant π (pi).

The setup involves two blocks of significantly different masses (here, 1 kg and 10000 kg) sliding on a frictionless surface. One block starts moving towards the other, and they collide elastically. The number of collisions before they come to rest or one block moves away is proportional to the digits of π.

For two blocks with masses in the ratio 1:100ⁿ, the number of collisions will be approximately equal to the first n+1 digits of π. In this case, with n = 3 (since 10000 = 100²), the number of collisions is close to the first 3 digits of π, which is 314.

This remarkable connection arises from the way the collisions and velocities behave, involving iterative calculations that mimic the series expansion of π.

How to get to 1000000 kg? and collisionCounter = 3141 ?

Since this is a computationally intensive during the collisions between left wall and 2 balls, perhaps slowing down the velocities of ball2 in x direction to -0.0001 might allow to compute the correct number of collisions before it becomes unstable? Good luck! and let me know if it works! Enjoy!

This remarkable connection arises from the way the collisions and velocities behave, involving iterative calculations that mimic the series expansion of π.

Research

[text]

Video

https://www.youtube.com/watch?v=6dTyOl1fmDo 

https://www.youtube.com/watch?v=vlUTlbZT4ig 

 Version:

1. https://weelookang.blogspot.com/2024/06/pi-collision-problem-simulator-finally.html

Other Resources

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Frequently Asked Questions: Colliding Blocks and Pi

1. What is the "colliding blocks" problem, and what makes it interesting? The colliding blocks problem involves two blocks on a frictionless plane, one initially stationary (smaller mass), and the other moving towards it (larger mass). The blocks undergo perfectly elastic collisions with each other and with a wall to the left of the smaller block. The puzzle lies in determining the total number of collisions that occur before both blocks move away from the wall. This problem is particularly fascinating because the total number of collisions is related to the digits of pi when the mass ratio of the larger block to the smaller block is a power of 100 (e.g., a ratio of 100:1 results in approximately 31 collisions, 10000:1 results in approximately 314, and so on).

2. How does the mass ratio of the blocks affect the number of collisions? The greater the mass ratio between the larger and smaller blocks, the more collisions will occur. A larger mass ratio means the more massive block has significantly more momentum. It takes many more collisions for the smaller block to effectively transfer enough momentum to the larger block to reverse its direction. As the mass ratio increases by factors of 100, the number of total collisions produces a sequence of numbers that begin with the digits of pi.

3. What are the key assumptions made in this idealized physics problem? To achieve the elegant connection to pi, several idealizing assumptions are made:

• Perfectly Elastic Collisions: No kinetic energy is lost during any collision (between the blocks or with the wall).
• Frictionless Plane: There is no friction between the blocks and the surface they are sliding on.
• Supermassive/Fixed Wall: The wall does not move or absorb any momentum from the collisions with the smaller block.
• Non-Relativistic Speeds: The speeds of the blocks are assumed to be much less than the speed of light, so relativistic effects are negligible, even for very large mass ratios where the smaller block experiences many rapid collisions.

4. How is the number of collisions related to pi in this system? The connection to pi arises when analyzing the problem in an abstract "velocity space." By rescaling the velocities based on the masses, the conservation of energy equation takes the form of a circle in this space. Each sequence of collisions (block-block, then block-wall) can be mapped to movements along chords and vertical lines within this circle. The total number of collisions corresponds to how many such moves are needed to reach an "end zone" in this velocity space, where both blocks are moving away from the wall. This number is determined by an angle related to the inverse tangent (arctangent) of the square root of the inverse mass ratio. For small angles (which correspond to large mass ratios), the arctangent of a small number is very close to the number itself. This approximation, combined with the geometry of the circle, leads to the number of collisions approximating the digits of pi.

5. Why do we see the digits of pi when the mass ratio increases by powers of 100? The mass ratio being a power of 100 is crucial because the slope in the rescaled velocity space is related to the square root of the mass ratio. For a mass ratio of \(100^n\), the square root is \(10^n\). The angle involved in the geometric interpretation is related to the arctangent of \(10^{-n}\). For large \(n\), \(\arctan(10^{-n}) \approx 10^{-n}\) radians. The number of "steps" or collisions needed to cover a sufficient portion of the circular state space is inversely proportional to this small angle. The way the digits of pi arise is analogous to asking how many times a small angle (approximately \(10^{-n}\) radians) fits into pi radians.

6. What is the connection between this classical mechanics problem and quantum computing (Grover's Algorithm)? While not fully explained in the provided text, the author mentions that the colliding blocks problem is "secretly connected to quantum computing," specifically Grover's Algorithm for search. The underlying mathematical structures and the way the system evolves through discrete steps bear a resemblance to the iterative nature of quantum algorithms like Grover's. The state space representation of the colliding blocks problem, where the system's state moves within a defined geometric boundary (the circle), has parallels in the state space of quantum systems and how quantum algorithms manipulate these states to find solutions. This connection highlights how seemingly unrelated areas of physics and computer science can share deep mathematical similarities.

7. Is the connection between the colliding blocks and pi a fully solved problem in mathematics? Technically, the full connection to pi is considered an unsolved problem. While the number of collisions closely approximates the digits of pi for mass ratios that are powers of 100, proving rigorously that this holds true without any off-by-one errors for arbitrarily large mass ratios is challenging. This difficulty arises from the reliance on small angle approximations and the fact that the precise nature of the digits of pi (specifically, the absence of arbitrarily long sequences of nines) cannot currently be proven mathematically.

8. Beyond the mathematical curiosity, why is studying this idealized problem valuable? Studying the colliding blocks problem, despite its idealizations, offers several valuable lessons. Firstly, it serves as an excellent example of problem-solving in physics, demonstrating the power of applying fundamental principles like conservation of energy and momentum, as well as the utility of transforming a problem into a more abstract mathematical or geometric representation. Secondly, it illustrates how seemingly simple physical systems can exhibit surprisingly complex and elegant connections to fundamental mathematical constants like pi. Finally, the hint at a connection to quantum computing underscores the idea that abstracting away real-world complexities can reveal hidden relationships between different scientific domains and pave the way for deeper understanding.