Reshape equation to match pattern

Question

I'm trying to solve some equations and i would like know if it is possible to have the Solve or Eliminate-function arrange the solution in a specific pattern.

In this example you can see that if you plug equations (2) and (3) into equation (1) and rearrange it a little bit, the first term becomes the kinetic energy and the second term is the power coefficient. I would like to tell Mathematica to include the kinetic energy term in the solution. I was thinking about something like this:

Eliminate[equations, {mDot, v2}, MatchPattern -> ERot == (1/2)*rho*A*v1^3*___]

Is there any way to make this possible? Any help or comment is greatly appreciated!

Answer 1

Clear["Global`*"]

eq1 = ERot == 1/2 m(v1^2 - v3^2);

eq2 = m == ρ A v2;

eq3 = v2 == (v1 + v3)/2;

To force use of the ratio v3/v1 use

eq4 = v31 == v3/v1;

Solve for ERot while eliminating {m, v2, v3}

(eq5 = (Solve[{eq1, eq2, eq3, eq4}, ERot, {m, v2, v3}][[1, 1]] // 
        Simplify) /.
      {a_*(b_ - 1) -> -a*(1 - b)} /.
     {v31 -> v3/v1, Rule -> Equal}) //
 TraditionalForm

(* ERot==1/4 A ρ v1^3 (1-v3/v1) (v3/v1+1)^2 *)

Checking,

eq1 === (Solve[{eq5, eq2, eq3}, ERot, {A, v2}][[1, 1]] /.
   Rule -> Equal)

(* True *)

To split the factor 1/4 you would need to Hold the subexpressions. You should check your result for the power coefficient.