How to draw a moving point with time?

Question

I am new to Mathematica. Try to draw a point moving with time. Like at t=0, a point is at p1(coordinates are random numbers); at t=1, this point is at p2. And there is always ONE point on the graph. Below is my code, xy[t_] is the point coordinate at time t in a 2D region.

I have two problems here,

1). Points can be out of this region. How to fix it?

2). All the points are shown on a same graph, how to show only one point at a certain time t?

Any help? Thank you.

xy[t_]=Module[{t},
         Table[
             If[t<1,{x[0]=0,y[0]=RandomReal[{-1,1}]},
                    {x[t]=RandomReal[{0,1}],y[t]=RandomReal[{-1,1}]}],{t,0,10}]]

Animate[Graphics[{Red,PointSize[0.05],Point[xy[t]]},
  AxesLabel->{"x","y"},
  PlotRange->{{0,10},{0,10}}],{t,0,10}]

Answer 1

Your creation of xy[t_]=... is not a good idea in the sense that xy[anything] produces a static table.

Try

xyPointList = 
 Module[{t}, 
  Table[If[t < 1, {x[0] = 0, 
     y[0] = RandomReal[{-1, 1}]}, {x[t] = 
      x[t - 1] + RandomReal[{0, 1}], 
     y[t] = y[t - 1] + RandomReal[{-1, 1}]}], {t, 0, 10}]]

(* {{0, 0.0251049}, {0.211352, 0.0367706}, {0.489274, 
  0.7281}, {1.39872, 0.538182}, {1.68534, 1.26225}, {2.18648, 
  1.66184}, {2.4956, 2.65684}, {2.69291, 1.87413}, {3.20496, 
  1.26491}, {3.30305, 2.02537}, {4.0114, 2.23029}} *)

Next use Map to produce a list of red points. I have adjusted the PlotRange so your points do not go off scale.

xyGraphicsList = 
  Graphics[{Red, PointSize[0.05], Point[#]}, AxesLabel -> {"x", "y"}, 
     PlotRange -> {{-1, 10}, {-3, 3}}] & /@ xyPointList;

Finally, use ListAnimate rather than Animate.

ListAnimate[xyList]

Answer 2

I am precomputing an Array of random numbers from 0 to 1 for x and -1 to 1 for y that will be the displacements. To that I Prepend {0,0} to define a fixed initial position. Then I Accumulate that list for a successive accumulated totals of elements. Then use Interpolation so the function is also defined at any arbitrary position between points.

xy = Interpolation[
   Accumulate[
    Prepend[
     Array[{RandomReal[{0, 1}], RandomReal[{-1, 1}]} &, 100],
     {0, 0}
     ]]
   , InterpolationOrder -> 1
   ];

Then the trajectory is like this

Plot[xy[t], {t, 0, 50}]

Now for the animation notice the {t, xy[t]} coordinates.

Animate[Graphics[{Red, PointSize[0.05], Point[{t, xy[t]}]}, 
  PlotRange -> {{0, 50}, {-5, 5}}], {t, 0, 
  50}]