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Relativity: Alice and Bob's Spaceship Paradox

Relativity: Alice and Bob's Spaceship Paradox
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2016-06-09 04:16
VectorCoordinate GeometryCalculusTransformationGeometric Construction
This example illustrates the Alice and Bob spaceship paradox in special relativity. Key mathematical concepts include relativity, Lorentz transformation, and Minkowski spacetime. Using GeoGebra commands such as Point, CurveCartesian, Intersect, etc., the Lorentz transformation can be visually demonstrated. Suitable for high school physics and mathematics courses to help students understand the basic principles of relativity.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. CurveCartesian((2 * sinh(k / 2)^(2)) + x(A), sinh(k), k, 0, 10) → Alice
  3. Point(xAxis) → B
  4. CurveCartesian((2 * sinh(k / 2)^(2)) + x(B), sinh(k), k, 0, 10) → Bob
  5. Intersect(HORIZON_{Alice}, xAxis, 1) → C_{Alice}
  6. Point(Alice) → X
  7. Tangent(X, Alice) → v_{Alice}
  8. Ray(C_{Alice}, X) → SIM_{Alice}
  9. Intersect(SIM_{Alice}, Bob) → SIM_{Alice, on Bob}
  10. Tangent(SIM_{Alice, on Bob}, Bob) → v_{Bob}
  11. Point(xAxis) → A_2
  12. CurveCartesian(((2 * sinh(k / 2)^(2)) * (x(A_2) - x(C_{Alice}))) + x(A_2), (sinh(k) * (x(A_2) - x(C_{Alice}))), k, 0, 10) → Alice_2
  13. Intersect(SIM_{Alice}, Alice_2) → SIM_{Alice, on Alice_2}
  14. Tangent(SIM_{Alice, on Alice_2}, Alice_2) → v_{Alice_2}
  15. Intersect(HORIZON_{Bob}, xAxis, 1) → C_{Bob}
  16. Segment(C_{Bob}, C_{Bob} + (1.5 * (SIM_{Alice, on Bob} - C_{Bob}))) → SIM_{Bob}
  17. Line(X, xAxis) → SIM_{LAB}
  18. Intersect(SIM_{LAB}, Alice_2) → SIM_{LAB, on Alice2}
  19. Intersect(SIM_{LAB}, Bob) → SIM_{LAB, on Bob}
  20. Tangent(SIM_{LAB, on Alice2}, Alice_2) → v_{Alice2,LAB}
  21. Tangent(SIM_{LAB, on Bob}, Bob) → v_{Bob,LAB}
  22. CurveCartesian(0, k, k, 0, 10) → LAB
  23. Intersect(SIM_{Bob}, Alice_2) → SIM_{Bob, on Alice2}
  24. Intersect(SIM_{Bob}, Alice) → SIM_{Bob, on Alice}
  25. Intersect(SIM_{LAB}, yAxis) → T_{LAB}
  26. Sequence(Segment[(-0.3, ((-k) * 0.3)) + X, (0.3, (k * 0.3)) + X], k, -1, 1, 2) → LC_X
  27. Sequence(Segment[(-0.3, ((-k) * 0.3)) + T_{LAB}, (0.3, (k * 0.3)) + T_{LAB}], k, -1, 1, 2) → LC_T
  28. Sequence(Segment[(-0.3, ((-k) * 0.3)) + SIM_{Alice, on Bob}, (0.3, (k * 0.3)) + SIM_{Alice, on Bob}], k, -1, 1, 2) → LC_S