kill(all)$
load(lrats)$
load(vect)$

Forces Airplane ( = pilot) , gliding flight and climb flight.'version 1.0'
peter.vlasschaert@gmail.com, 04/09/2018

see : introduction to flight ( = John D Anderson  Jr)  used by flight engineers.
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use : soft ( Geogebra 5 : Figures)
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general :  four forces :  D  = drag (  friction force)
                                            L   = lift
                                            W  = Weight  ( mass under gravitation : m*g, g= constant)
                                            T   =  thrust
ref  :  Horizontal  ( = flight direction in general)
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S I (units ) : g = 9.81 m/s^2                             (acceleration)
                      forces = Newton = kg*m/s^2    (force              )
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forces :  1e) a long the flight path                          ,  vector  flight  ( = ref),positiv
                 2e) perpendicular ( to the flight path)   , vector  L         ( = ref),positiv

very important : L perpendicular to D  ( angle = π /2) o f 90° degree
                               W opposite L
                               D opposite  T
plane :                  point  mass

p1:W= m*g;

general ( = Horizontal flight ) :ref
===========================

  horizontal flight

1e)  a long the flight path (= flight) ,   ( = forces)
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p2: T -D =0;
p2a :solve(p2,T);
p2b:p2a[1];

2e) perpendicular to flight path (= flight) , (= forces)
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p3: L -W=0;
p3a :solve(p3,L);
p3b:p3a[1];
p3c:subst([p1],p3b);

Gliding flight  (  Horizontal = ref ), angle : θ  = α
=======================================

 forces Gliding flight (ref : pos direction : orange)

;

1e)  a long the flight path (= flight) ,   ( = forces)
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 => vertical projection of the forces

p4:D-W*sin(α) =0;
p4a :solve(p4,D);
p4b:p4a[1];
p4c:subst([p1,α=θ],p4b);

2e) perpendicular to flight path (= flight) , (= forces)
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 => vertical projection of the forces

p5:L-W*cos(α) =0;
p5a :solve(p5,L);
p5b:p5a[1];
p5c:subst([p1,α=θ],p5b);

find : tan(θ) => θ ' p4c,p5c'
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p5d: p4c/p5c;

p5d1:lratsubst([rhs(p5d) =tan(θ)],p5d);

p5d2:solve(p5d1,θ);

p5d3:p5d2[1];

rem : no  T : by Gliding flight

Climb flight  (  Horizontal = ref ), angle : θ  = α
=======================================

forces of climbing flight

1e)  a long the flight path (= flight) ,   ( = forces)
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 => vertical projection of the forces

p6:T-D-W*sin(α) =0;
p6a :solve(p6,T);
p6b:p6a[1];
p6c:subst([p1,α=θ],p6b);

2e) perpendicular to flight path (= flight) , (= forces)
-----------------------------------------------------------------------------
 => vertical projection of the forces

p7:L-W*cos(α) =0;
p7a :solve(p7,L);
p7b:p7a[1];
p7c:subst([p1,α=θ],p7b);

find : tan(θ) => θ ' p6c,p7c'
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p7d:p6c-D;

p7d1:p7c;

p7d2:p7d/p7d1;

p7d3:lratsubst([rhs(p7d2) =tan(θ)],p7d2);

p7d4:solve(p7d3,θ);

p7d5:p7d4[1];

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