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相对论：爱丽丝与鲍勃的飞船悖论

相对论：爱丽丝与鲍勃的飞船悖论
几何画板及绘图内容加载中，请稍等……
2016-06-09 04:16
向量解析几何微积分变换几何作图
这个范例演示了相对论中的爱丽丝与鲍勃的飞船悖论。核心数学概念包括相对论、洛伦兹变换和时空图。通过使用GeoGebra命令如Point、CurveCartesian、Intersect等，可以构建出洛伦兹变换的图形。适合高中物理和数学课程，帮助学生理解相对论的基本原理。

GeoGebra 命令及参数

  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