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differential equation applications 227The frictional force F (f) is not a basic force of nature but rather a simple modelof a complicated phenomenon. We do know that friction always opposes motion,which means it is in the direction opposite to velocity. One model assumes that thefrictional force is proportional to a power n of the projectile’s speed [M&T 03]:F (f) = −km|v| nv|v| , (9.74)where the −v/|v| factor ensures that the frictional force is always in a directionopposite that of the velocity. Physical measurements indicate that the power n isnoninteger and varies with velocity, and so a more accurate model would be anumerical one that uses the empirical velocity dependence n(v). With a constantpower law for friction, the equations of motion ared 2 xdt 2 = −kvn xv x|v| ,d 2 ydt 2 = −g − kvn yv y|v| ,|v| = √v 2 x + v 2 y. (9.75)We shall consider three values for n, each of which represents a different model forthe air resistance: (1) n =1for low velocities; (2) n = 3 2, for medium velocities; and(3) n =2for high velocities.9.16.1 Simultaneous Second-Order ODEsEven though (9.75) are simultaneous second-order ODEs, we can still use ourregular ODE solver on them after expressing them in standard formdydtWe pick y to be the 4-D vector of dependent variables:= y(t, y) (standard form). (9.76)y (0) = x(t),y (1) = dxdt , y(2) = y(t), y (3) = dydt . (9.77)We express the equations of motion in terms of y to obtain the standard form:dy (0)dtdy (2)dt(≡ dx )= y (1) ,dt(≡ dy )= y (3) ,dtdy (1)dtdy (3)dt( )≡ d2 xdt 2= 1 m F x(f) (y)( )≡ d2 ydt 2 = 1 m F y (f) (y) − g.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 227