Suppose that an airplanre departs from the point (200,0) onthe xy-plane and main

Question

Suppose that an airplanre departs from the point (200,0) onthe xy-plane and maintains a heading toward an airport at theorigin. The plane travels with constant speed v = 500 mi/h relativeto the wind, which is blowing due north (along y-axis) withconstant speed w = 50 mi/h. Find the trajectory y = f(x) of theplane. use differential equ to solve Suppose that an airplanre departs from the point (200,0) onthe xy-plane and maintains a heading toward an airport at theorigin. The plane travels with constant speed v = 500 mi/h relativeto the wind, which is blowing due north (along y-axis) withconstant speed w = 50 mi/h. Find the trajectory y = f(x) of theplane. use differential equ to solve use differential equ to solve

Explanation / Answer

Since the plane is always flyingdirectly toward the airport we can make an angleq at the originwhich points at the plane. The wind has no effect on the xcomponent of the plane's velocity which is -vcos(q),the minus being necessary because the plane is flying in thedirection of decreasing x as the time t advances.If w is the speed of the wind, then the plane's ycomponent of velocity is w-vsin(q).Then the following equations must be true:    x'[t] = -vcos() = -(vx)/(x2+y2), and    y'[t] = w -v sin() = w - (vy)/(x2+y2) =(w(x2+y2) -vy)/(x2+y2) Divide the second equation by the first toeliminate the dependence upon t and you obtain:    y'[x] = (v y[x]- w(x2+y[x]2))/(vx) This differential equation can be solvedwith the initial condition that y[200] = 0 using any methodyou know to get:    y[x] = xsinh((log(200)-log(x))/10)    x'[t] = -vcos() = -(vx)/(x2+y2), and    y'[t] = w -v sin() = w - (vy)/(x2+y2) =(w(x2+y2) -vy)/(x2+y2) Divide the second equation by the first toeliminate the dependence upon t and you obtain:    y'[x] = (v y[x]- w(x2+y[x]2))/(vx) This differential equation can be solvedwith the initial condition that y[200] = 0 using any methodyou know to get:    y[x] = xsinh((log(200)-log(x))/10)

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