Plotting a moving ball for projectile motion with animate

Question

Here is my code for plotting an orbit. What I want to do is have the "planet" shown as it orbits in the simulation. Can anyone help with this?

Clear["Global`"];
Off[General::spell, General::spell1];
dt = 0.01;
vx = 0.0;
vy = 1.2;
x = 1.0;
y = 0.0;
r2 = x*x + y*y;
r = Sqrt[r2];
f = -1/r2;
fx = x/r*f;
fy = y/r*f;
Clear[orbit];
orbit = {};
Do[{vx = vx + fx*dt, vy = vy + fy*dt, x = x + dt*vx, y = y + dt*vy, 
   r2 = x^2 + y^2, r = Sqrt[r2], f = -1/r2, fx = x/r*f, fy = y/r*f, 
   AppendTo[orbit, {x, y}]}, {i, 1, 2000}];
orbit1 = Table[orbit[[i]], {i, 1, Length[orbit]}];
Animate[ListPlot[orbit1[[1 ;; j]], 
  PlotRange -> {{-3, 3}, {-2.5, 2.5}}], {j, 1, Length[orbit], 1}]

Answer 1

The same, but more Mathematica-ish (working with functions, representing vectors as lists and using NestList instead of variables and looping-appending):

Clear["Global`"];
dt = .01;
velInit = {0, 1.2};
posInit = {1.0, 0};
f[pos_] := -pos/(pos.pos)^(3/2);
vel[{velP_, posP_}] := velP + f[posP] dt
pos[{velP_, posP_}] := posP + vel[{velP, posP}] dt

orbit = NestList[{vel@#, pos@#} &, {velInit, posInit}, 1500][[All, 2]];

Animate[ListPlot[orbit, Joined -> True, 
                Epilog -> {PointSize@.05, Purple, Point[orbit[[j]]]}], 
{j, 1, 1500, 1}]

Edit

You may experiment with different initial conditions:

orbitFun[{velInit_, posInit_}] := NestList[{vel@#, pos@#} &, {velInit, posInit}, 1500][[All, 2]]

Manipulate[ DynamicWrapper[  
                 Column[{Grid[{{"Intial Velocity", "Initial Position"}, {velInit,  posInit}}], 
                 Animate[ListPlot[orb, Joined -> True, 
                         Epilog -> {PointSize@.05, Purple, Point[orb[[j]]]}], {j, 1, 1500,1}]}],
                 orb = orbitFun[{velInit, posInit}]],
           {velInit, {.1, .1}, {1.2, 1.2}}, {posInit, {-2, -2}, {2, 2}}]

Edit

Code dissection

(* Your formula for "f" *)
f[pos_] := -pos/(pos.pos)^(3/2);
(* New vel based on old vel and old pos *)
vel[{velP_, posP_}] := velP + f[posP] dt
(*New pos based on old vel and old pos *)
pos[{velP_, posP_}] := posP + vel[{velP, posP}] dt

(* Now the tricky NetsList[] *)
(* It will recurse 1500 times storing in the results list the {vel, pos} for each iteration *)
(* Each itertion get feeded with the last iteration {vel, pos} and calculate the new one*)
(*AFTER it finishes, the [[All,2]] thingy will keep only the positions 
and discard the velocities so that we can plot the pos list*)

orbit = NestList[{vel@#, pos@#} &, {velInit, posInit}, 1500][[All, 2]];