Summary of Solutions and Terminal Velocity for Linear and Quadratic Models

Explicit formulae for the velocity and position can be obtained using dsolve . While the dsolve command could be used, I have been unable to force Maple to present the solution to the system as nicely as I am able to obtain by first solving for the velocity and then the position. The acceleration and jerk can also be expressed directly in terms of the velocity. The theoretical terminal velocity is also easily obtained. Note that the velocity should be negative. It might be possible to use the assume facility to impose conditions that help Maple simplify some of the results, but I prefer to avoid using assume if possible.

> i := 'i':

> for i in [lin,quad] do

>

> if i=lin then printf("\n\nLINEAR MODEL\n============\n")

> elif i=quad then printf("\n\nQUADRATIC MODEL\n===============\n")

> fi;

>

> # S := dsolve( EOM[i] union IC, {x(t),v(t)} );

> # X[i] := collect( allvalues( subs( S, x(t) ) ), {m,g,k} );

> # V[i] := normal( allvalues( subs( S, v(t) ) ) );

> V[i] := rhs( normal( allvalues( dsolve( { ode(EOM[i],v), ic(IC,v(0)) }, v(t) ) ) ) );

> X[i] := rhs( collect( dsolve( { subs( v(t)=V[i], ode(EOM[i],x) ), ic(IC,x(0)) }, x(t) ), {m,g,k} ) );

> A[i] := collect( simplify( subs( {v(t)=V[i], x(t)=X[i]}, solve( ode(EOM[i],v), diff(v(t),t) ) ) ), {m,g} );

> J[i] := subs( solve(ode(EOM[i],v),{diff(v(t),t)}), diff(subs(k=k(t),A[i]),t) );

>

> ACC[i] := rhs( op( solve( subs({diff(v(t),t)=a(t),k=k(t)}, ode(EOM[i],v)), {a(t)} ) ) );

> JERK[i] := collect( subs( diff(v(t),t)=a(t), diff( ACC[i], t ) ), {m,g,k(t),diff(k(t),t)} );

>

> Vterm[i] := rhs( ode(EOM[i],v) ) = 0;

>

> printf("\nTerminal Velocity\n" );

> print( v[T]=solve(Vterm[i],v(t)) );

>

> printf("\nSolutions: expressed in terms of position and velocity" );

> print( a=ACC[i] );

> print( j=JERK[i] );

>

> printf("\nSolutions: expressed explicitly in terms of time" );

> print( x=X[i] );

> print( v=V[i] );

> print( a=A[i] );

> print( j=J[i] );

> od:

LINEAR MODEL

============

Terminal Velocity

Solutions: expressed in terms of position and velocity

Solutions: expressed explicitly in terms of time

QUADRATIC MODEL

===============

Terminal Velocity

Solutions: expressed in terms of position and velocity

Solutions: expressed explicitly in terms of time

>