There's more to those colliding blocks that compute pi

Key Concepts

• Elastic Collisions: Collisions where no kinetic energy is lost.
• Conservation of Energy: The total energy of an isolated system remains constant.
• Conservation of Momentum: The total momentum of an isolated system remains constant.
• State Space: A space in which all possible states of a system are represented, with each possible state corresponding to one unique point.
• Inscribed Angle Theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended by it at any point on the remaining part of the circle.
• Small Angle Approximation: For small angles, sin(θ) ≈ θ, tan(θ) ≈ θ, and arctan(θ) ≈ θ.
• Grover's Algorithm: A quantum algorithm for searching an unsorted database quadratically faster than classical algorithms.

Block Collision Problem and Pi

• The Setup: Two blocks on a frictionless plane, one stationary (small mass, m2), the other moving towards it (large mass, m1). A wall is to the left of the blocks. The goal is to determine the total number of collisions (including wall collisions).
• Collision Count and Mass Ratio: The number of collisions is related to the mass ratio (m1/m2). As the mass ratio increases by powers of 100, the number of collisions approximates the digits of pi.
  • m1/m2 = 1: 3 collisions
  • m1/m2 = 100: 31 collisions
  • m1/m2 = 10,000: 314 collisions
  • m1/m2 = 1,000,000: 3,141 collisions
• Idealizations: The puzzle relies on idealizations like perfectly elastic collisions (no energy loss) and ignoring relativistic effects for very large mass ratios.
• Real-World Demonstrations: While idealized, the phenomenon can be observed for smaller mass ratios (e.g., 100:1).

Problem-Solving Approach

• Relevant Laws: Conservation of energy and conservation of momentum are key.
  • Energy: 1/2 * m1 * v1^2 + 1/2 * m2 * v2^2 = constant
  • Momentum: m1 * v1 + m2 * v2 = constant
• State Space: A coordinate plane is created where the x-coordinate represents v1 (velocity of the large block) and the y-coordinate represents v2 (velocity of the small block). The point (v1, v2) moves within this space as the blocks collide.
• Energy Conservation and Ellipse: The conservation of energy equation defines an ellipse in the state space. The point (v1, v2) is constrained to move along this ellipse.
• Rescaling Coordinates for Symmetry: To simplify the problem and relate it to circles (and thus pi), the coordinates are rescaled:
  • x = sqrt(m1) * v1
  • y = sqrt(m2) * v2
  • This transforms the ellipse into a circle: x^2 + y^2 = constant
• Momentum Conservation and Line: In the rescaled coordinates, the conservation of momentum equation becomes a linear equation, representing a line in the state space. The slope of this line is -sqrt(m1/m2).
• Collision as Intersection: After a collision, the point (x, y) moves to the other intersection point of the momentum line and the energy circle.
• Wall Collision: A wall collision flips the sign of the y-coordinate (v2), changing the momentum and shifting the momentum line.
• End Zone: The experiment ends when both blocks are moving to the right (x > 0, y > 0) and the small block is slower than the large block (v2 < v1). This defines a region in the state space called the "end zone."

Geometric Interpretation and Pi

• Circle Puzzle: The physics problem is transformed into a geometric puzzle: Start at the leftmost point of a circle, move down and to the right along a line with slope -sqrt(m1/m2), then straight up to the circle, then down and to the right again, and so on, until you reach the "end zone." The puzzle is to count the number of lines drawn.
• Equal Arcs: The arcs between the points where the lines intersect the circle are all equal in length. This can be proven using the inscribed angle theorem.
• Angle Theta: The angle between the vertical line and the down-and-to-the-right line is denoted as theta. Each arc covers an angle of 2*theta.
• Number of Arcs and Circumference: The number of arcs that can be dropped before reaching the end zone is determined by how many times 2theta can be added to itself before exceeding the total circumference of the circle (2pi).
• Tangent and Arctangent: The tangent of theta is equal to sqrt(m2/m1). Therefore, theta = arctan(sqrt(m2/m1)).
• Small Angle Approximation: For small angles, arctan(x) ≈ x. This approximation is used to relate the angle theta to a power of 10 when the mass ratio is a power of 100.
• Digits of Pi: The small angle approximation explains why the number of lines drawn in the circle diagram (and thus the number of collisions) has the same digits as pi.

Unsolved Problem

• Small Angle Approximation Error: The small angle approximation introduces a small error.
• Consecutive Nines in Pi: If the digits of pi ever contain a sequence where the next n digits after the first n digits are all nines, the collision count could be off by one.
• Unproven Conjecture: Proving that such a sequence of nines does not exist in the digits of pi is beyond the current capabilities of mathematics, making the connection between colliding blocks and pi technically an unsolved problem.

Broader Implications

• General Problem-Solving: The block collision problem is a good example of how to approach complex problems by:
  • Listing relevant equations and theorems.
  • Drawing pictures and using visual intuition.
  • Respecting symmetries.
  • Simplifying the problem to its core essence.
• Hidden Connections: The problem highlights the importance of abstraction in mathematics and physics, as it can reveal hidden connections between seemingly unrelated concepts.
• Quantum Computing Connection: The block collision problem is secretly connected to Grover's algorithm in quantum computing, which will be explored in a subsequent video.
• Analogy to Light Beams: The colliding blocks are analogous to a beam of light bouncing between two mirrors at an angle.