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Re: Projectile Launched Around the Earth

• To: mathgroup at smc.vnet.net
• Subject: [mg116431] Re: Projectile Launched Around the Earth
• From: "Ted Ersek" <ersekt at md.metrocast.net>
• Date: Tue, 15 Feb 2011 06:33:23 -0500 (EST)

The code below is n improved and more readable version of the demonstration
at

 <http://demonstrations.wolfram.com/ProjectileLaunchedAroundTheEarth/>
http://demonstrations.wolfram.com/ProjectileLaunchedAroundTheEarth/

(*-----------------------------------------------------------------------*)



Manipulate[Module[{DiffEquations,ProjectilePosition,ProjectilePath,

     EarthRadius,ProjectileCrashed,CrashSite,Projectile,Earth},

     DiffEquations={D[x[t],t,t] == -4*Pi^2*x[t]*(x[t]^2+y[t]^2)^(-3/2),

            D[y[t],t,t]==-4*Pi^2*y[t]*(x[t]^2+y[t]^2)^(-3/2),

            x[0]==0,y[0]==y0,(x=92[0]==Vx0,(y=92[0]==0};

      With[{DiffEqnSolution=NDSolve[DiffEquations, {x,y}, {t,0,t2}]},

           ProjectilePosition[t_Real]={x[t],y[t]}/.DiffEqnSolution;

           ProjectilePath=ParametricPlot[ ProjectilePosition[t], {t,0,t2},
PerformanceGoal->"Quality"];

           ProjectileCrashed=MemberQ[ ProjectilePath, {_,_}?(Norm[#]<1&),
{6} ];

           If[ProjectileCrashed,

                ProjectilePath=ProjectilePath/.Line[pnts_]:>Line[ TakeWhile[
pnts, (1<Norm[#]&) ]];

                CrashSite=Cases[ ProjectilePath,Line[pnts_]:>Last[pnts],-1];

                Projectile=Graphics[{Red, PointSize[0.02], Point[CrashSite]
}],

             

               (* else *)

               Projectile=Graphics[{Red, PointSize[0.02],
Point[ProjectilePosition[t2]] }]

           ];

           Earth=Graphics[{Gray,Disk[{0,0},1]}];

           Show[ Earth, ProjectilePath, Projectile, PlotRange->4,
Axes->False] ]

      ],

      {{t2, 2.9, "t"}, 0.01, 7.75},

      {{y0, 1.4, "Subscript[y, 0]"}, 1.1, 3.97},

      {{Vx0, 6.2, "Subscript[v, Subscript[x, 0]]"}, 1, 6.75}

]





(*--------------------------------------------------------------------------
------------------------------------*)

(*------How do we modify the code above to correctly model traveling around
the Earth?  ----*)

When I tried to do it  NDSolve  complained that the equations were probably
stiff, and would not work it out automatically.

Note:

   G = 6.672*^-11 (Newton (m/kg)^2)

   EarthMass =  5.9721986=D710^24 kg

   EarthRadius = 6378.14 km

    F = G*EarthMass*ProjectileMass*(DistanceFromEarthCenter)^-2



Please indicate the units used for (t2, y0, Vx0)



Note:

EarthRadius above is the actual equatorial radius of the earth.

For simplicity we can model the earth as

a Sphere with(radius) = (actual equatorial radius).

I will ignore all complications such as wind resistance, variations in
density of the earth, etc.



Thanks,

    Ted Ersek