Sure. First, the height of the bounce was controlled by the -.9 factor you see in the velocity of the rebound. (Physically, the energy at the top is mgh, and at the bounce is 1/2 mv^2, so the relationship is not linear.) We can make the height the same by using -1 for the y velocity change. As to what we draw, the solution just gives us functions for the location. This can be used for any graphic, with Animate or Export used to produce animations. The attached notebook demonstrates this.

Of course, this just moves the object, which still bounces like a ball. It would be more difficult to cause the object shape to be involved in detecting the hit on the surface, and far more difficult to include it in the physics. For example, by really simulating the linear and rotational motions of a cube made of elastic material bouncing chaotically.

Best, David

Attachments:

q = {{x[t]}, {y[t]}};

dq = D[q, t];
ddq = D[dq, t];


dx = D[x, t];
dy = D[y, t];
ddx = D[dx, t];
ddy = D[dy, t];


KE = 0.5 m (x'[t]^2 + y'[t]^2);
PE = m*g*y[t];

L = KE - PE;

(*The constraint*)
\[Phi] = y;
d\[Phi] = D[\[Phi], q];

Eq = (Thread[D[D[L, dq\[Transpose]], t] - D[L, q\[Transpose]] == 0]);

ELtemp = Solve[Eq[[1]] && Eq[[2]], Flatten[ddq]];

EL = {x''[t] == ELtemp[[1, 1, 2]], y''[t] == ELtemp[[1, 2, 2]]};

p = D[L, dq\[Transpose]];
H = p.dq - L;

m = 1;
g = 9.8;

ICnt = 5;
xzero = 0; xdzero = 3; yzero = 10; ydzero = 0; Tzero = 0;
x1 = {}; y1 = {}; Time = {}; P1 = {}; Hset = {};

(*The calculations of the Lagrangian just before and after impact \
with the wall*)
FullSimplify[D[L, dq]*dq - L];
(*This is the condition just before impact and is constrained to hit \
the wall along x=0*)
FullSimplify[
  eq1 = (Thread[(D[L, dq] /. t -> Tplus) - (D[L, dq] /. 
         t -> Tminus) == \[Lambda]*d\[Phi]])];
(*This is the condition just after inpact and there is no constraint*)


FullSimplify[
  eq2 = (Thread[((D[L, dq]*dq - L) /. 
         t -> Tplus) - ((D[L, dq]*dq - L) /. t -> Tminus) == 0])];
ImpactTemp = 
  Solve[eq1[[1]] && eq2[[1]], 
   Join[{x'[Tplus]}, {y'[Tplus]}, {\[Lambda]}]];

For[i = 0, i < ICnt, i++, 
  InitCon = {\[Theta][Tzero] == xzero && 
     yzero, \[Theta]'[Tzero] == -xdzero && -ydzero};
  sol = NDSolve[Join[EL, Initial], {x, y}, {t, Tzero, 30}, 
    Method -> {"EventLocator", "Event" -> y, 
      "EventAction" :> 
       Throw[{Tend = t, yend = y[t], ydotend = y'[t]}, 
        "StopIntegration"]}];
  xsol = x[t] /. sol[[1]]; ysol = y[t] /. sol[[1]]; 
  Hsol = H /. sol[[1]];
  x1 = Append[x1, xsol]; y1 = Append[y1, ysol]; 
  Time = Append[Time, Tend];
  Hset = Append[Hset, {Hsol, Tzero < t <= Tend}]; 
  P1 = Append[P1, {xsol && ysol, Tzero < t <= Tend}];
  yzero = yend; ydzero = ydotend;
  Tzero = Tend
  ];
    \[Theta]finalsol = Piecewise[P1];
    Hfinalsol = Piecewise[Hset];



(*InitCon={x'[0]\[Equal]3, x[0]==0, y[0]==10, y'[0]==0};


sol=NDSolve[Join[EL,InitCon],{x,y},{t,0,30}] ;
xsol=x[t]/.sol[[1]];
ysol=y[t]/.sol[[1]];*)

(*PLOTS*)
Plot[xsol, {t, 0, 2}, AxesLabel -> {t, x}]
Plot[ysol, {t, 0, 2}, AxesLabel -> {t, y}]
ParametricPlot[{xsol, ysol}, {t, 0, 2}, AxesLabel -> {x, y}]

(*Hamiltonian and its Plot*)
Plot[\[Theta]finalsol, {t, 0, Tend}, 
 AxesLabel -> {t, HoldForm[\[Theta]]}]
Plot[Hfinalsol, {t, 0, Tend}, AxesLabel -> {t, HoldForm[H]}]

(*Plot[xsol,{t,0,10}, AxesLabel\[Rule]{t,x}]
Plot[ysol,{t,0,10}, AxesLabel\[Rule]{t,y}]*)
X[\[Tau]_] := R*Sin[\[Theta]finalsol] /. t -> \[Tau];
Y[\[Tau]_] := -R*Cos[\[Theta]finalsol] /. t -> \[Tau];


(*ANIMATION*)
X[\[Tau]_] := xsol /. \[Theta]finalsol /. t -> \[Tau];
Y[\[Tau]_] := ysol /. \[Theta]finalsol /. t -> \[Tau];

Animate[Graphics[{PointSize[0.05], Point[{ X[\[Tau]], Y[\[Tau]]}], 
   Line[{{-10, 0}, {10, 0}}], Line[{{0, -10}, {0, 10}}]}], {\[Tau], 0,
   2}]

I tried this to get a ball to bounce to 5 times and then stop. But it gives me certain errors like equations are not a quantified system of equalities and some replacement rules are not valid.

Any suggestions on how to proceed?

Thanks Varun

Attachments:

Works perfectly.

Say if I want to use Event Locator[] like the one I have used for the pendulum and solve an impact equation, do i use it inside Manipulate?