This is the region here the force of gravity is greater than the combined lift and centrifugal forces. If your plane/satellite is in this region for too long, you will not go to space today.
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Expression 27: "v" equals "x"v=x
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this is just defining the x axis as v for velocity. it makes the equations look nice.
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Expression 29: "p" equals "d" Subscript, "k" , Baseline left parenthesis, "y" , right parenthesisp=dky
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Expression 30: "d" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis equals left brace, "x" less than or equal to "b" Subscript, "t" "k" , Baseline left bracket, 1 , right bracket : StartFraction, "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis Over "r" Subscript, "k" , Baseline "t" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis left bracket, 1 , right bracket , EndFraction , "x" less than or equal to "b" Subscript, "t" "k" , Baseline left bracket, 2 , right bracket : StartFraction, "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis Over "r" Subscript, "k" , Baseline "t" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis left bracket, 2 , right bracket , EndFraction , "x" less than or equal to "b" Subscript, "t" "k" , Baseline left bracket, 3 , right bracket : StartFraction, "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis Over "r" Subscript, "k" , Baseline "t" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis left bracket, 3 , right bracket , EndFraction , "x" less than or equal to "b" Subscript, "t" "k" , Baseline left bracket, 4 , right bracket : StartFraction, "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis Over "r" Subscript, "k" , Baseline "t" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis left bracket, 4 , right bracket , EndFraction , "x" less than or equal to "b" Subscript, "t" "k" , Baseline left bracket, 5 , right bracket : StartFraction, "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis Over "r" Subscript, "k" , Baseline "t" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis left bracket, 5 , right bracket , EndFraction , "x" less than or equal to "b" Subscript, "t" "k" , Baseline left bracket, 6 , right bracket : StartFraction, "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis Over "r" Subscript, "k" , Baseline "t" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis left bracket, 6 , right bracket , EndFraction , "x" less than or equal to "b" Subscript, "t" "k" , Baseline left bracket, 7 , right bracket : StartFraction, "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis Over "r" Subscript, "k" , Baseline "t" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis left bracket, 7 , right bracket , EndFraction , "x" less than or equal to "b" Subscript, "t" "k" , Baseline left bracket, 8 , right bracket : StartFraction, "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis Over "r" Subscript, "k" , Baseline "t" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis left bracket, 8 , right bracket , EndFraction , "x" greater than or equal to 70000 : 0 , right bracedkx=x≤btk1:pkxrktkx1,x≤btk2:pkxrktkx2,x≤btk3:pkxrktkx3,x≤btk4:pkxrktkx4,x≤btk5:pkxrktkx5,x≤btk6:pkxrktkx6,x≤btk7:pkxrktkx7,x≤btk8:pkxrktkx8,x≥70000:0
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Expression 31: "t" equals StartFraction, "x" minus "a" Over "b" minus "a" , EndFractiont=x−ab−a
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Expression 32: "h" Subscript, 00 , Baseline left parenthesis, "x" , right parenthesis equals left parenthesis, 1 plus 2 "x" , right parenthesis left parenthesis, 1 minus "x" , right parenthesis squaredh00x=1+2x1−x2
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Expression 33: "h" Subscript, 01 , Baseline left parenthesis, "x" , right parenthesis equals "x" squared left parenthesis, 3 minus 2 "x" , right parenthesish01x=x23−2x
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Expression 34: "h" Subscript, 10 , Baseline left parenthesis, "x" , right parenthesis equals "x" left parenthesis, 1 minus "x" , right parenthesis squaredh10x=x1−x2
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Expression 35: "h" Subscript, 11 , Baseline left parenthesis, "x" , right parenthesis equals "x" squared left parenthesis, "x" minus 1 , right parenthesish11x=x2x−1
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Expression 36: "p" Subscript, "k" , Baseline left parenthesis, "x" , right parenthesis equals "h" Subscript, 00 , Baseline left parenthesis, "t" , right parenthesis "p" Subscript, 1 , Baseline plus "h" Subscript, 10 , Baseline left parenthesis, "t" , right parenthesis left parenthesis, "b" minus "a" , right parenthesis "m" Subscript, 1 , Baseline plus "h" Subscript, 01 , Baseline left parenthesis, "t" , right parenthesis "p" Subscript, 2 , Baseline plus "h" Subscript, 11 , Baseline left parenthesis, "t" , right parenthesis left parenthesis, "b" minus "a" , right parenthesis "m" Subscript, 2 , Baseline left brace, "a" less than or equal to "x" less than or equal to "b" , right bracepkx=h00tp1+h10tb−am1+h01tp2+h11tb−am2a≤x≤b