In the projectile problem, when air resistance is taken into account but the var

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In the projectile problem, when air resistance is taken into account but the variation of gravitational force with altitude is neglected, the governing differential equation and initial conditions are m d2x2/dt2 + k dx2/dt2 = -mg, X2(0) = 0, dx2dt2(0) = V0, where air resistance exerts a force proportional to the speed of the projectile with proportionality constant k > 0 and acts in a direction to resist motion. When the effect of air resistance is small, the change of variables is still appropriate (why does this follow from the concepts of scaling and introduction of nondimentional variables ?). Assume an expansion of the form for the solution of the nondimentionalized version of E0(1). proceed as in the corresponding problem treated in class to find tm and xm to O[B]2. Also compute T, the time it takes the projectile to return to earth, and explain why Check the method of 1 by solving the problem exactly for V(X), finding Xm, and expanding it as a series in B for B "small".

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