Billiard Balls 

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X GraphicsGrid[{Map[ ParametricPlot[{x[t], y[t]} /. sol1, {t, 0, #}, Frame -> True, FrameTicks -> None, PlotRange -> 1, Axes -> False, PlotStyle -> Blue] &, {5, 20, 100}], Map[ParametricPlot[{x[t], y[t]} /. sol2, {t, 0, #}, Frame -> True, FrameTicks -> None, PlotRange -> 1, Axes -> False, PlotStyle -> Red] &, {5, 20, 100}]}]

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X GraphicsGrid[{Map[ ParametricPlot3D[{x[t], y[t], z[t]} /. sol3d1, {t, 0, #}, Axes -> False, PlotStyle -> Blue] &, {5, 20, 100}], Map[ParametricPlot3D[{x[t], y[t], z[t]} /. sol3d2, {t, 0, #}, Axes -> False, PlotStyle -> Red] &, {1, 50, 200}]}]

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Model the trajectory of a billiard ball struck from two nearly identical angles.

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X {sol1, sol2} = Map[NDSolve[{x'[t] == a[t], y'[t] == b[t], x[0] == 0, y[0] == 0, a[0] == 1, b[0] == #, WhenEvent[x[t]^2 == 1, a[t] -> -a[t]], WhenEvent[y[t]^2 == 1, b[t] -> -b[t]]}, {x, y}, {t, 0, 100}, DiscreteVariables -> {a, b}] &, {Rationalize[Sqrt[2], .01], Sqrt[ 2]}];

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Model the trajectory of a 3D billiard ball struck from two different angles.

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X ball3d[r1_?NumericQ, r2_?NumericQ] := NDSolve[{x'[t] == a[t], y'[t] == b[t], z'[t] == c[t], x[0] == 0, y[0] == 0, z[0] == 0, a[0] == 1, b[0] == r1, c[0] == r2, WhenEvent[x[t]^2 - 1, a[t] -> -a[t]], WhenEvent[y[t]^2 - 1, b[t] -> -b[t]], WhenEvent[z[t]^2 - 1, c[t] -> -c[t]]}, {x[t], y[t], z[t]}, {t, 0, 200}, DiscreteVariables -> {a, b, c}];

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X {sol3d1, sol3d2} = {ball3d[1/3, 2/3], ball3d[Sqrt[2], Sqrt[3]]};

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