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Parabolas in Sports: Tennis Service

Parabolas in Sports: Tennis Service
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2020-02-02 16:57
VectorFunctionCoordinate GeometryGeometric Construction
This example demonstrates the parabolic trajectory of a tennis ball during service, highlighting key mathematical concepts such as parabolas, coordinate geometry, vectors, and function graphing. It leverages GeoGebra commands like Line, Point, Segment, OrthogonalLine, Intersect, Polygon, Midpoint, Rotate, Angle, Function, Vector, Translate, If, Mirror, and CircumcircleArc to construct and analyze the parabolic path and related geometric elements. Suitable for high school students learning about parabolas and their applications in physical motion, or for mathematical modeling education.

GeoGebra Commands & Parameters

  1. Line(S, xAxis) → n_1
  2. Line(F_1, xAxis) → m
  3. Line(M, m) → k
  4. Point(m) → L
  5. Line(S, I_1) → t
  6. Point(t) → D_1
  7. Line(I_1, xAxis) → s
  8. Point(xAxis) → E_1
  9. Segment(Service, G) → a
  10. Point(a) → F
  11. OrthogonalLine(F, a) → i
  12. Point(i) → hautfilet
  13. Segment(F, hautfilet) → e
  14. Line(L, e) → n
  15. Line(F_1, n) → b_1
  16. Intersect(s, n) → C_1
  17. Polygon(D_1, E_1, C_1, I_1) → poly1, d_1, e_1, c_1, i_1
  18. Point(e_1) → K_1
  19. Midpoint(K_1, C_1) → S_1
  20. Midpoint(S_1, C_1) → U_1
  21. Line(U_1 + (0, 0.2), n_1) → q_1
  22. Line(K_1, q_1) → s_1
  23. Intersect(s_1, n) → A_2
  24. Line(K_1, k) → a_1
  25. Intersect(a_1, b_1) → P_1
  26. Intersect(n, k) → N
  27. Intersect(f, i) → Ballhaute, A
  28. Intersect(f, yAxis) → ball
  29. Line(ball, xAxis) → b
  30. Intersect(f, b) → B, E
  31. Midpoint(E, ball) → J
  32. Midpoint(ball, J) → H
  33. Intersect(f, xAxis) → C, Impact
  34. Rotate(E, α, ball) → E'
  35. Angle(E, ball, E') → β
  36. Segment(Service, G) → c
  37. Point(xAxis) → fond
  38. Segment(G, fond) → d
  39. Intersect(e, xAxis) → D
  40. Line(hautfilet, D) → g
  41. Function(f, 0, x(Impact)) → h
  42. Rotate(H, α, ball) → J'
  43. Vector(ball, J') → u
  44. Point(xAxis) → O
  45. Segment(fond, Service) → j
  46. Segment(W, Z) → l
  47. Translate(W, (x(Impact) / 18.5 * Vector[W, Z])) → W'
  48. If(y(Ballhaute) ≟ y(hautfilet), 1, If[y(Ballhaute) < y(hautfilet), 2, 0]) → mm
  49. Function(f, 0, x(hautfilet)) → p
  50. Segment(W, W') → q
  51. Segment(R, Q) → r
  52. Line(E_1, n) → f_1
  53. Intersect(f_1, a_1) → L_1
  54. Intersect(k, f_1) → M_1
  55. Intersect(f_1, m) → N_1
  56. Intersect(k, b_1) → O_1
  57. Polygon(N, A_2, P_1, M) → poly3, p_1, t_1, a_2, b_2
  58. Polygon(F_1, L, N, O_1) → poly2, g_1, h_1, j_1, k_1
  59. Polygon(N, A_2, P_1, M) → poly4, l_1, m_1, c_2, d_2
  60. Midpoint(d_1) → Q_1
  61. Midpoint(c_1) → R_1
  62. Midpoint(i_1) → T_1
  63. Midpoint(Q_1, R_1) → V_1
  64. Intersect(n_1, e_1) → W_1
  65. Midpoint(S, W_1) → Z_1
  66. Line(Q_1, R_1) → r_1
  67. Segment(Z_1, V_1) → e_2
  68. Intersect(r_1, q_1) → B_2
  69. Intersect(i_1, q_1) → C_2
  70. Line(B_2, xAxis) → f_2
  71. Intersect(e_1, f_2) → D_2
  72. Segment(C_2, D_2) → g_2
  73. Segment(V_1, B_2) → h_2
  74. Segment(W_1, S) → i_2
  75. Midpoint(Q_1, E_1) → E_2
  76. Midpoint(E_2, E_1) → F_2
  77. Midpoint(D_1, Q_1) → G_2
  78. Midpoint(D_1, G_2) → H_2
  79. Segment(H_2, J_1) → k_2
  80. Mirror(H_2, J_1) → I_2
  81. Mirror(J_1, J_1) → J_2
  82. Segment(I_2, J_2) → l_2
  83. Mirror(J_1, R_1) → K_2
  84. Segment(F_2, K_2) → j_2
  85. Intersect(g_2, k_2) → L_2
  86. Intersect(i_2, k_2) → M_2
  87. Intersect(j_2, i_2) → N_2
  88. Intersect(g_2, j_2) → O_2
  89. Segment(Z_1, N_2) → m_2
  90. Segment(Z_1, M_2) → n_2
  91. Segment(L_2, O_2) → p_2
  92. CircumcircleArc(Q, R, P_2) → q_2
  93. Segment(Q_2, R_2) → r_2
  94. Intersect(l, r_2) → S_2
  95. Segment(Q, P_2) → s_2
  96. Translate(Q, (x(Impact) / 18.5 * Vector[Q, P_2])) → Q'
  97. CircumcircleArc(Q', R, Q) → t_2
  98. Midpoint(q) → T_2
  99. Vector(W, W') → v
  100. Vector(Q, Q') → w
  101. Translate(W, (Animation / x(Impact) * v)) → W'_1
  102. Segment(W, W'_1) → a_3
  103. Translate(Q, (Animation / x(Impact) * w)) → Q'_1