Thanks David, that worked perfectly! My final question is now that i have our value for t* at r=5.2, how do i find o'(t*) (the first derivative of o at our value of t)?

Cheers! Mike

This actually works fantastically! how do i manipulate the code to instead give me the value of o at our given rlimit? This was my attempt, and it failed horribly!

rLimit = 5.2; tList = {}; oList = {};

{fo, fr} = 
  NDSolveValue[{r''[t] == - (4*Pi^2 / r[t]^2) + ((o'[t]^2)*r[t]) + 
      HeavisideTheta[5 - t]*(0.51*Cos[x]/((1 - A) + A (1 - B (t)))), 
    o''[t] == (-2 r'[t]*o'[t])/r[t] + 
      HeavisideTheta[
        5 - t]*(( 0.51*Sin[x])/(r[t]*((1 - A) + A (1 - B (t))))), 
    o[0] == 0, o'[0] == 2*Pi, r[0] == 1, r'[0] == 0, 
    WhenEvent[
     r[t] >= rLimit, {AppendTo[tList, {t, r[t]}], 
      AppendTo[oList, {o, r[o]}]}]}, {o[t], r[t]}, {t, -10, 20}];

Regards

Mike

Hi Mike,

Let's not stop the integration -- we'll just accumulate a list of the positive crossings:

In[157]:= L = 2 Pi;
M = 1;
x = 35 Degree;

In[160]:= rLimit = 2.5; tList = {};

In[161]:= (* notice I used NDSolveValue to get o[t] and r[t] *)
{fo, fr} = 
  NDSolveValue[{r''[t] == -(4*Pi^2/r[t]^2) + ((o'[t]^2)*r[t]) + 
      HeavisideTheta[5 - t]*0.51*Cos[x], 
    o''[t] == (-2 r'[t]*o'[t])/r[t] + 
      HeavisideTheta[5 - t]*0.51*Sin[x], o[0] == 0, o'[0] == 2*Pi, 
    r[0] == 1, r'[0] == 0, 
    WhenEvent[r[t] >= rLimit, AppendTo[tList, {t, r[t]}]]}, {o[t], 
    r[t]}, {t, -10, 10}];

In[162]:= (* here are the positive crossings *)
tList

Out[162]= {{5.27189, 2.5}, {8.97694, 2.5}}

In[163]:= (* The first is of course at t = *)
tList[[1, 1]]

Out[163]= 5.27189

In[164]:= (* We can see them on the plot *)
Plot[fr, {t, -10, 10}, 
 Epilog -> {Red, PointSize[Large], Point /@ tList}]

Regarding my earlier comment, I think tStop, or in this case tList, needs to be defined before its use in WhenEvent because that provides it with global scope. Otherwise it has scope local to NDSolve and expires when NDSolve finishes.

Best Regards, David

Trying that, it returns the error FindRoot::nlnum: The function value {-5.3+<<1>>[t]} is not a list of numbers with dimensions {1} at {x} = {7.}. >>, i looked up the syntax and help documents but i couldn't get it to give me anything except that error. I assume its to do with the First function as it is treating it like a list?

Well actually, scratch that, 15.22 is a correct solution, however it is not the first solution on the graph, by which i mean

Is there any way to specify the first solution at around t=7

Hello again!

Sorry about the disagreement with the graph, in my currently used model, different initial conditions are set such that r does eventually reach the desired value.

What is the syntax of the Nsolve and Findroot when solving a specific equation like First[r[t] /. TotalThrust] == 5.3?

I've attempted with NSolve [First[r[t] /. TotalThrust] == 5.3, t] and it returns a value of 16.22 which is wrong when looking at my graph, and returns the error : Inverse functions are being used by NSolve, so some solutions may not be found; use Reduce for complete solution information. >>

Thanks!