积极 AI Logo 积极 AI

球的跳跃与反弹运动模拟

球的跳跃与反弹运动模拟
几何画板及绘图内容加载中，请稍等……
2020-02-01 16:35
函数数列微积分参数方程滑块动画
① 本范例通过GeoGebra动态模拟一个球在重力作用下的跳跃与反弹过程，直观展示物体运动轨迹与速度变化。② 核心数学概念包括函数图像、数列（反弹序列）、线性方程（速度变化）、微分（瞬时速度）以及概率（随机因素可能影响反弹高度），同时涉及缩放（Dilation）与指数衰减（反弹高度递减）。③ 作图要点：使用Vector定义初速度，Intersect求交点，If条件判断反弹，CurveCartesian绘制曲线轨迹，Segment和Line构造参考线，Sequence生成多次反弹序列，Polygon构建辅助图形；对象包括多个点（E, D, F）、参数（t, v, u）、函数（k, h, g）及系列反弹参数（r1~r10, a1~a10, t1~t10）。④ 适合高中至大学低年级学生，用于物理与数学结合的运动学教学，也可用于微积分中导数概念的直观演示。

GeoGebra 命令及参数

  1. Vector((0, 1)) → v
  2. Vector((1, 0)) → u
  3. Intersect(xAxis, yAxis) → O
  4. If(t ≥ 0 ∧ t ≤ r1, ((k_1 * (t + r1)) * (t - r1)), 0) → a1
  5. If(t ≥ 0 ∧ t ≤ r1, t, 0) → t1
  6. If(t ≥ r1 ∧ t ≤ r2, ((k_1 * (t - r1)) * (t - r2)), 0) → a2
  7. If(t ≥ r1 ∧ t ≤ r2, t, 0) → t2
  8. If(t ≥ r2 ∧ t ≤ r3, ((k_1 * (t - r2)) * (t - r3)), 0) → a3
  9. If(t ≥ r2 ∧ t ≤ r3, t, 0) → t3
  10. If(t ≥ r3 ∧ t ≤ r4, ((k_1 * (t - r3)) * (t - r4)), 0) → a4
  11. If(t ≥ r3 ∧ t ≤ r4, t, 0) → t4
  12. If(t ≥ r4 ∧ t ≤ r5, ((k_1 * (t - r4)) * (t - r5)), 0) → a5
  13. If(t ≥ r4 ∧ t ≤ r5, t, 0) → t5
  14. If(t ≥ r5 ∧ t ≤ r6, ((k_1 * (t - r5)) * (t - r6)), 0) → a6
  15. If(t ≥ r5 ∧ t ≤ r6, t, 0) → t6
  16. If(t ≥ r6 ∧ t ≤ r7, ((k_1 * (t - r6)) * (t - r7)), 0) → a7
  17. If(t ≥ r6 ∧ t ≤ r7, t, 0) → t7
  18. If(t ≥ r7 ∧ t ≤ r8, ((k_1 * (t - r7)) * (t - r8)), 0) → a8
  19. If(t ≥ r7 ∧ t ≤ r8, t, 0) → t8
  20. If(t ≥ r8 ∧ t ≤ r9, ((k_1 * (t - r8)) * (t - r9)), 0) → a9
  21. If(t ≥ r8 ∧ t ≤ r9, t, 0) → t9
  22. If(t ≥ r9 ∧ t ≤ r10, ((k_1 * (t - r9)) * (t - r10)), 0) → a10
  23. If(t ≥ r9 ∧ t ≤ r10, t, 0) → t10
  24. If(t ≥ r10 ∧ t ≤ r11, ((k_1 * (t - r10)) * (t - r11)), 0) → a11
  25. If(t ≥ r10 ∧ t ≤ r11, t, 0) → t11
  26. If(t ≥ r11 ∧ t ≤ r12, ((k_1 * (t - r11)) * (t - r12)), 0) → a12
  27. If(t ≥ r11 ∧ t ≤ r12, t, 0) → t12
  28. If(t ≥ r13 ∧ t ≤ r14, ((k_1 * (t - r13)) * (t - r14)), 0) → a14
  29. If(t ≥ r13 ∧ t ≤ r14, t, 0) → t14
  30. If(t ≥ r14 ∧ t ≤ r15, ((k_1 * (t - r14)) * (t - r15)), 0) → a15
  31. If(t ≥ r14 ∧ t ≤ r15, t, 0) → t15
  32. If(t ≥ r15 ∧ t ≤ r16, ((k_1 * (t - r15)) * (t - r16)), 0) → a16
  33. If(t ≥ r15 ∧ t ≤ r16, t, 0) → t16
  34. If(t ≥ r16 ∧ t ≤ r17, ((k_1 * (t - r16)) * (t - r17)), 0) → a17
  35. If(t ≥ r16 ∧ t ≤ r17, t, 0) → t17
  36. If(t ≥ r17 ∧ t ≤ r18, ((k_1 * (t - r17)) * (t - r18)), 0) → a18
  37. If(t ≥ r17 ∧ t ≤ r18, t, 0) → t18
  38. If(t ≥ r18 ∧ t ≤ r19, ((k_1 * (t - r18)) * (t - r19)), 0) → a19
  39. If(t ≥ r18 ∧ t ≤ r19, t, 0) → t19
  40. If(t ≥ r19 ∧ t ≤ r20, ((k_1 * (t - r19)) * (t - r20)), 0) → a20
  41. If(t ≥ r19 ∧ t ≤ r20, t, 0) → t20
  42. If(t ≥ r12 ∧ t ≤ r13, ((k_1 * (t - r12)) * (t - r13)), 0) → a13
  43. If(t ≥ r12 ∧ t ≤ r13, t, 0) → t13
  44. CurveCartesian(s, ((k_1 * (s + r1)) * (s - r1)), s, 0, r1) → c1
  45. CurveCartesian(s, ((k_1 * (s - r1)) * (s - r2)), s, r1, r2) → c2
  46. CurveCartesian(s, ((k_1 * (s - r2)) * (s - r3)), s, r2, r3) → c3
  47. CurveCartesian(s, ((k_1 * (s - r3)) * (s - r4)), s, r3, r4) → c4
  48. CurveCartesian(s, ((k_1 * (s - r4)) * (s - r5)), s, r4, r5) → c5
  49. CurveCartesian(s, ((k_1 * (s - r5)) * (s - r6)), s, r5, r6) → c6
  50. CurveCartesian(s, ((k_1 * (s - r6)) * (s - r7)), s, r6, r7) → c7
  51. CurveCartesian(s, ((k_1 * (s - r7)) * (s - r8)), s, r7, r8) → c8
  52. CurveCartesian(s, ((k_1 * (s - r8)) * (s - r9)), s, r8, r9) → c9
  53. CurveCartesian(s, ((k_1 * (s - r9)) * (s - r10)), s, r9, r10) → c10
  54. CurveCartesian(s, ((k_1 * (s - r10)) * (s - r11)), s, r10, r11) → c11
  55. CurveCartesian(s, ((k_1 * (s - r11)) * (s - r12)), s, r11, r12) → c12
  56. CurveCartesian(s, ((k_1 * (s - r12)) * (s - r13)), s, r12, r13) → c13
  57. If(t ≤ r20, P1, P2) → P
  58. CurveCartesian(s, ((k_1 * (s - r13)) * (s - r14)), s, r13, r14) → c14
  59. CurveCartesian(s, ((k_1 * (s - r14)) * (s - r15)), s, r14, r15) → c15
  60. CurveCartesian(s, ((k_1 * (s - r15)) * (s - r16)), s, r15, r16) → c16
  61. CurveCartesian(s, ((k_1 * (s - r16)) * (s - r17)), s, r16, r17) → c17
  62. CurveCartesian(s, ((k_1 * (s - r17)) * (s - r18)), s, r17, r18) → c18
  63. CurveCartesian(s, ((k_1 * (s - r18)) * (s - r19)), s, r18, r19) → c19
  64. CurveCartesian(s, ((k_1 * (s - r19)) * (s - r20)), s, r19, r20) → c20
  65. Segment(AA, (-1, 0)) → c
  66. Segment((-2, 0), (-2, h)) → d
  67. Line((0, h), c) → e
  68. Intersect(e, d) → A
  69. Sequence(AA + (i / 5 * Vector[AA, BB]), i, 0, 5) → liste1
  70. Polygon(B, C, BB, AA) → poly1, i, j, l, m
  71. Polygon(D, E, F, G) → poly2, d_1, e_1, f_1, g_1
  72. Segment(D1, (0, F1)) → G1
  73. Segment(D2, F2) → G2
  74. Segment(D3, F3) → G3
  75. Segment(D4, F4) → G4
  76. Segment(D5, F5) → G5
  77. Segment(D6, F6) → G6
  78. Segment(D7, F7) → G7
  79. Segment(D8, F8) → G8
  80. Segment(D9, F9) → G9
  81. Segment(D10, F10) → G10
  82. Segment(D11, F11) → G11
  83. Segment(D12, F12) → G12
  84. Segment(D13, F13) → G13
  85. Segment(D14, F14) → G14
  86. Segment(D15, F15) → G15
  87. Segment(D16, F16) → G16
  88. Segment(D17, F17) → G17
  89. Segment(D18, F18) → G18
  90. Segment(D19, F19) → G19
  91. Segment(D20, F20) → G20
  92. Segment(H1, H2) → I1
  93. Segment(H2, H3) → I2
  94. Segment(H3, H4) → I3
  95. Segment(H4, H5) → I4
  96. Segment(H5, H6) → I5
  97. Segment(H6, H7) → I6
  98. Segment(H7, H8) → I7
  99. Segment(H8, H9) → I8
  100. Segment(H9, H10) → I9
  101. Segment(H10, H11) → I10
  102. Segment(H11, H12) → I11
  103. Segment(H12, H13) → I12
  104. Segment(H13, H14) → I13
  105. Segment(H14, H15) → I14
  106. Segment(H15, H16) → I15
  107. Segment(H16, H17) → I16
  108. Segment(H17, H18) → I17
  109. Segment(H18, H19) → I18
  110. Segment(H19, H20) → I19
  111. Sequence(Text[If[i ≟ 2, "ar", "ar^{" + (i - 1) + "}"], Element[liste2, i] + (-0.2, 0.4), true, true], i, 1, Length[liste2] - 10) → liste3

AI 绘图提示语参考

画一个动态模拟小球在重力作用下跳跃和反弹的图形，展示运动轨迹和速度变化。图形应包含初始速度向量、坐标轴交点、多个反弹高度递减的曲线、参考线以及表示多个反弹的辅助形状。使用参数控制小球运动，并用函数图展示速度变化，同时用序列生成多次反弹，适合演示微积分中的导数概念。
以上提示语为根据原始绘图命令整理而来，仅供参考。