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Homework 5, Problem Number 8 A woman bails out of an airplane at an altitude of ten thousand feet, falls freely for twenty seconds , then opens her parachute. Assume linear air resistance v ft/s2 , taking = 0:15 without the parachute and = 1:5 with the parachute. Suggestion: First determine her height above the ground and velocity when the parachute opens. First, observe that the woman's velocity must satisfy the di erential equation dv = g v: dt (1) This is a rst order linear di erential equation. Therefore, by writing dv + v = g (2) dt we see that we have P (t) = and Q(t) = g. Multiplying both sides of this equation by the integrating factor (t) = exp yields et Equivelantly, this gives Z dt = et ; dv + et v = get : dt et v 0 = get : Therefore, we see that e v= g t and Z v(t) = et dt = g + Ce (3) (4) (5) g t e + C; t (6) (7) is the general solution to (1). To determine the woman's velocity when the parachute opens, we need to solve for C using the initial condition v(0) = 0. This gives, 0 = g + C; (8) so the woman's velocity at time t before she opens her parachute is given by v(t) = g g + e where g = 32 and = 0:15. 1 t ; (9) Therefore, at time t = 20, her velocity is v(20) = 202:712 feet per second. Notice that in general we will have g C = v(0) + : (10) To nd her height above the ground when the parachute opens, rst notice that y(t) = Z Z g g t + e dt + C1 = g t g2 e t + C1 : v(t)dt + C1 = (11) Because y(0) = 10000, we see that 10000 = g2 + C1 ; so the woman's height above the ground at time t is given by y(t) = g g t+ e 2 t + 10000 + g2 (12) (13) Therefore, at time t = 20, her height is y(20) = 7084:75 feet. Notice that in general we will have v(0) g C1 = y(0) + (14) + 2 To determine how long it will take her to reach the ground, we will need to solve a very similar problem, using the velocity and height just computed as new initial conditions. Therefore, the woman's velocity at time t seconds after opening her parachute is the solution to the initial value problem dv = g v; v(0) = 202:712: dt (15) Just as before, the general solution to this di erential equation is given by (7). Therefore, g g (16) v(t) = t + v(0) + e t : Similarly, her position is given by v(0) g g e + y(t) = t 2 = 21:3333t + 120:919e t g + y(0) + v(0) + 2 1 :5 t + 6964:83 Setting y(t) = 0 and solving for t (numerically) gives t = 326:476. 2 (17)