% 1
clear all;
M=5.9722e+024;        %    (.)
R=6371008.8;          %     (.)
G = 6.6740831e-11;    %   (3 -1 -2)
g=G*M/(R^2)           %   
% 2
%    
T=10 
% 3    g
t=T;
S=eval('(g*t^2)/2')
% 4   
clear R;
clear t;
syms h v1 K R t
hs=dsolve('Dh=t*K/(R+h)^2', 'h(0)=0')
% 5 
R=6371008.8;
K=G*M
t=T;
k=1;
for k=1:length(hs)
  F=hs(k);
  F=simplify(F);
  s=eval(F);
  if isreal(s) 
    F
    s
    delta=S-s
    break;
  end;
end;
% ----------------------------------------
% 2-
%  
% h=1/2*(12*t^2*G*M+8*R^3)^(1/3)-R

clear all
syms K R x
taylor(1/2*(12*x^2*K+8*R^3)^(1/3)-R,3,0)
F='1/2*8^(1/3)*(R^3)^(1/3)-R+1/4*8^(1/3)*(R^3)^(1/3)*K/R^3*x^2';

% 
clear all
M=5.9722e+024;        %    (.)
R=6371008.8;          %     (.)
G = 6.6740831e-11;    %   (3 -1 -2)
g=G*M/(R^2) 
K=G*M
x=10
h0=(g*x^2)/2
h1=1/2*8^(1/3)*(R^3)^(1/3)-R+1/4*8^(1/3)*(R^3)^(1/3)*K/R^3*x^2


% 
clear all 
syms v K R
F = 'v*x - (1/2*8^(1/3)*(R^3)^(1/3)-R+1/4*8^(1/3)*(R^3)^(1/3)*K/R^3*x^2) = 0'
x=solve(F)
% 
x= 2*v*R^2/K

% 
clear all;
M=5.9722e+024;        %    (.)
R=6371008.8;          %     (.)
G = 6.6740831e-11;    %   (3 -1 -2)
g=G*M/(R^2)           %   

v=100
t1=2*v/g
K=G*M
t2=2*v*R^2/K