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Re: NDSolve and InterpolatingFunction
*To*: mathgroup at smc.vnet.net
*Subject*: [mg56148] Re: NDSolve and InterpolatingFunction
*From*: rknapp at wolfram.com
*Date*: Sat, 16 Apr 2005 03:53:36 -0400 (EDT)
*References*: <d3o06i$blo$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Virgil Stokes wrote:
> I am solving the following system of ODE's
>
> g = 9.81 ; (* acceleration of gravity [m/s^2] *)
> d = 0.063; (* diameter of ball [m] *)
> m = 0.05; (* mass in kg *)
> \[Rho] = 1.29; (* air density, [kg/m^3] *)
> \[Alpha] = \[Rho] Pi d^2/(8 m)
>
> h = 1; (* initial height [m] *)
> v0 = 25; (* magnitude of ball velocity [m/s] *)
> \[Theta] = 15 (* initial angle of release [15 degrees] *)
>
> vars = {x[t], vx[t], z[t], vz[t]}
>
> initc = {x[0] == 0, vx[0] == v0*Cos[\[Theta] Degree], z[0] == h,
> vz[0] == v0*Sin[\[Theta] Degree]}
>
> v[t] = Sqrt[vx[t]^2 + vz[t]^2];
>
> eq1 = x'[t] == vx[t];
> eq2 = vx'[t] == -0.508 \[Alpha] vx[t] v[t];
> eq3 = z'[t] == vz[t];
> eq4 = vz'[t] == -g - 0.508 \[Alpha] vz[t] v[t];
> eqns = {eq1, eq2, eq3, eq4}
>
> sol = NDSolve[Join[eqns, initc], vars, {t, 0, 25}]
>
> which works fine; but how can I find (e.g. using Solve) the value of
t
> such that z[t] is 0; i.e, where, z[t] (in the
> form of an InterpolatingFunction) is zero.
>
You won't be able to do it with Solve since the result in sol is a
numerical approximation, not a symbolic function.
FindRoot, however works just fine after running the commands you gave
above,
In[12]:=
FindRoot[z[t] /. sol, {t,0,25}]
Out[12]=
{t->1.36201}
In version 5.1, an even more convenient way to get the point at which z
is zero is to use the EventLocator method:
In[13]:=
sol=NDSolve[Join[eqns,initc],vars,{t,0,25},
Method->{"EventLocator", "Event"->z[t], "EventAction":>(tzroot = t)}];
tzroot
Out[13]=
1.36201
You will find documentation for EventLocator in the NDSolve
AdvancedDocumentation:
http://documents.wolfram.com/mathematica/Built-inFunctions/AdvancedDocumentation/DifferentialEquations/NDSolve/ODEIntegrationMethods/ControllerMethods/EventLocation.html
Rob Knapp
Wolfram Research
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