Question

You bail out of a helicopter and pull the ripcord of your parachute. Now the air resistance proportionality constant is k 1.65, so your downward velocity satisfies the initial va ft/s and t in seconds. In order to investigate your chances of survival, construct a slope field for this differential equation and sketch the appropriate solution curve. What will located haystack do any good? How long will it take you to reach 95% of your limiting velocity 32 1.65v, v(0) 0 Construct a slope field for this differential equation and sketch the appropriate solution curve. Choose the correct graph below 10 0.0 0.2 0.6 0.8 0.0 0.2 0.4 0.6 0.8

Answer 1

It is given that the downward velocity of a helicopter satisfies the following initial value problem dv/dt = 32 - 1.6v v(0) = 0 The objective is to construct a slope field for this differential equation and to sketch the appropriate solution curve. Also, need to find the limiting velocity and to find that how long it will take to reach 95 % of our limiting velocity. Now consider the differential equation du/dt = 32 - 1.6v This implies: dv/dt + 1.6v = 32 That is a linear equation in u. Compare This equation with dv/dt + Pu = Q This implies P = 1.6 Q = 32. \r\n Now, integrating factor is: I.F = e^integral P middot dx = e^integral 1.6 dx I.F = e^1.6x Solution is: u(I.F) = integral Q(I.F)dt This implies u e^1.6t = integral 32 e^1.6 t dt = 32 e^1.6x /1.6 + c