Maple worksheet to validate perturbation approximation to 'tm' (time to maximum height) for projectile problem. ~/maple/courses/applied-math/projectile.mw C.G. (10-03) original version C.G. (05-07) .mw version Re-initialize workspace. restart ; Parameters (in English units, miles per hour, etc): gravitational constant, radius of the earth. g := 78500 ;

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R := 4000 ;

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Formulas for true (integral) and approximate (algebraic) solutions. tmTrue := epsilon -> 1/sqrt(2*epsilon) * int( 1/sqrt(epsilon/2-eta/(1+eta)), eta=0..epsilon/(2-epsilon) ) ;

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tmApprox := epsilon -> 1 + 2/3*epsilon + 2/5*epsilon^2 ;

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Compare true vs approximate formulas for the 3 test velocities. for V in [ 100, 1000, 10000 ] do 'V' = V ; epsilon := evalf(V^2/g/R) ; True := tmTrue(epsilon) ; Approx := tmApprox(epsilon) ; Error := True - Approx ; end do ;

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Plot true vs approximate formulas on same graph for epsilon parameter between 0 and 1. plot( [ tmTrue, tmApprox ], 0..1, legend=[ "True", "Approx" ], linestyle=[ SOLID, DASH ], axes=boxed ) ;

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