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Five-Ball Newton’s Cradle: Symmetric Collisions and Geometric Trajectories

Five-Ball Newton’s Cradle: Symmetric Collisions and Geometric Trajectories
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2020-08-19 04:52
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This example demonstrates the motion of a five-sphere Newton’s cradle, showing how identical free balls exchange momentum through repeated collisions. By adjusting initial pull settings (e.g., angle and length-related parameters), the construction reveals how symmetry and consistent geometric relationships shape the observed trajectories. Key mathematics includes circle/arc geometry, angle construction and measurement, correspondence of positions under symmetry-like updates, and parameterized control of motion via angular variables (such as θ and φ) to drive dynamic positions. In GeoGebra, the model is built using Line and Point for the pendulum lines and pivot-related points, Angle to define the pulling angle, Intersect to compute key geometric intersections, Polygon for outlines, and If for conditional updates of states. For the 3D presentation, Cylinder and Sphere render the physical components, while CircumcircleArc helps establish circular/arc structure. Dynamic object parameters and controllers (e.g., lengthController, widthController, heightController, pendulumRadius, trajectoryRadius, g) together with objects like pullPoint_i, restPosition, trajectoryCenter_i, and dynamicPosition_i generate and update each ball’s position and trajectory. Suitable for upper secondary through early university instruction on geometric constructions, parameterized animation, and mathematical visualization of physical-style systems.

GeoGebra Commands & Parameters

  1. Line(basePoint, zAxis) → j
  2. Point(j) → heightController
  3. Line(basePoint, yAxis) → g
  4. Point(g) → widthController
  5. Line(basePoint, xAxis) → f
  6. Point(f) → lengthController
  7. CircumcircleArc(trajectoryCenter_{3} + ((-pullRadius), 0, 0), trajectoryCenter_{3} + (0, 0, (-pullRadius)), trajectoryCenter_{3} + (pullRadius, 0, 0)) → e_2
  8. Point(e_2) → pullPoint_{3}
  9. Angle(pullPoint_{3}, trajectoryCenter_{3}, trajectoryCenter_{3} + restPosition, yAxis) → θ_{i}
  10. CircumcircleArc(trajectoryCenter_{4} + ((-pullRadius), 0, 0), trajectoryCenter_{4} + (0, 0, (-pullRadius)), trajectoryCenter_{4} + (pullRadius, 0, 0)) → r_2
  11. Point(r_2) → pullPoint_{4}
  12. Angle(pullPoint_{4}, trajectoryCenter_{4}, trajectoryCenter_{4} + restPosition, yAxis) → φ_{i}
  13. CircumcircleArc(trajectoryCenter_{2} + ((-pullRadius), 0, 0), trajectoryCenter_{2} + (0, 0, (-pullRadius)), trajectoryCenter_{2} + (pullRadius, 0, 0)) → d_2
  14. Point(d_2) → pullPoint_{2}
  15. CircumcircleArc(trajectoryCenter_{1} + ((-pullRadius), 0, 0), trajectoryCenter_{1} + (0, 0, (-pullRadius)), trajectoryCenter_{1} + (pullRadius, 0, 0)) → c_2
  16. Point(c_2) → pullPoint_{1}
  17. Angle(pullPoint_{1}, trajectoryCenter_{1}, trajectoryCenter_{1} + restPosition, yAxis) → γ_{i}
  18. CircumcircleArc(trajectoryCenter_{5} + ((-pullRadius), 0, 0), trajectoryCenter_{5} + (0, 0, (-pullRadius)), trajectoryCenter_{5} + (pullRadius, 0, 0)) → s_2
  19. Point(s_2) → pullPoint_{5}
  20. Angle(pullPoint_{5}, trajectoryCenter_{5}, trajectoryCenter_{5} + restPosition, yAxis) → ω_{i}
  21. Angle(pullPoint_{2}, trajectoryCenter_{2}, trajectoryCenter_{2} + restPosition, yAxis) → δ_{i}
  22. Line(widthController, f) → h
  23. Line(lengthController, g) → i
  24. Intersect(i, h) → D
  25. Polygon(basePoint, lengthController, D, widthController) → q1, a, b, d, c
  26. Cylinder(J, K, cylinderRadius) → e, k, p, l
  27. Cylinder(M, L, cylinderRadius) → m, q, r, n
  28. Cylinder(H, J, cylinderRadius) → o, s, t, u
  29. Cylinder(F, K, cylinderRadius) → v, c_1, d_1, w
  30. Cylinder(E, L, cylinderRadius) → a_1, e_1, f_1, b_1
  31. Cylinder(G, M, cylinderRadius) → g_1, g_2, g_3, g_4
  32. Cylinder(F, E, cylinderRadius) → h_1, k_1, p_1, i_1
  33. Cylinder(G, H, cylinderRadius) → j_1, q_1, r_1, l_1
  34. If(time ≤ π / 2, trajectoryCenter_{1} + dynamicPosition_{1}, trajectoryCenter_{1} + dynamicPosition_{5}) → sphereCenter_{1}
  35. Segment(U, sphereCenter_{1}) → f_2
  36. Segment(sphereCenter_{1}, V) → h_2
  37. If(time ≤ π / 2, trajectoryCenter_{2} + dynamicPosition_{2}, trajectoryCenter_{2} + dynamicPosition_{4}) → sphereCenter_{2}
  38. Segment(Z, sphereCenter_{2}) → i_2
  39. Segment(sphereCenter_{2}, W) → j_2
  40. Segment(sphereCenter_{3}, A_1) → k_2
  41. Segment(sphereCenter_{3}, B_1) → l_2
  42. If(time ≤ π / 2, trajectoryCenter_{4} + dynamicPosition_{4}, trajectoryCenter_{4} + dynamicPosition_{2}) → sphereCenter_{4}
  43. Segment(D_1, sphereCenter_{4}) → m_2
  44. Segment(sphereCenter_{4}, C_1) → n_2
  45. If(time ≤ π / 2, trajectoryCenter_{5} + dynamicPosition_{5}, trajectoryCenter_{5} + dynamicPosition_{1}) → sphereCenter_{5}
  46. Segment(E_1, sphereCenter_{5}) → p_2
  47. Segment(sphereCenter_{5}, F_1) → q_2
  48. Sphere(sphereCenter_{1}, sphereRadius) → m_1
  49. Sphere(sphereCenter_{2}, sphereRadius) → n_1
  50. Sphere(sphereCenter_{3}, sphereRadius) → o_1
  51. Sphere(sphereCenter_{4}, sphereRadius) → s_1
  52. Sphere(sphereCenter_{5}, sphereRadius) → t_1

AI Drawing Prompt Reference

Draw a diagram of a five-ball Newton's cradle showing symmetric collisions and geometric trajectories. The diagram should include five ball bodies connected to symmetric trajectory centers, with paths illustrating post-collision motion. Angles and parameters should be reflected in the layout. Construction order: first establish trajectory centers, then draw the balls, and finally connect them to the centers. Maintain symmetry to highlight geometric relationships and dynamic trajectories.
The prompt above was compiled from the original drawing commands and is for reference only.