Problem 5 (20 points): The figure below shows the forces exerted on a hot air balloon system. dy FD Figure Forces on a h

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Problem 5 20 Points The Figure Below Shows The Forces Exerted On A Hot Air Balloon System Dy Fd Figure Forces On A H 1 (254.35 KiB) Viewed 39 times

Problem 5 (20 points): The figure below shows the forces exerted on a hot air balloon system. dy FD Figure Forces on a hot air balloon: FB = buoyancy, FG = weight of gas, Fp = weight of payload (including the balloon envelope), and FD = drag. Note that the direction of the drag is downward when the balloon is rising. = Formulate the drag force as Fp = pav?ACą, where pa = air density (kg/mº), v= velocity (m/s), Pa v2 A = projected frontal area (m²), and Ca = the dimensionless drag coefficient = 0.47 for a sphere). Note also that the total mass of the balloon consists of two components: m = mg + mp, where mg = the mass of the gas inside the expanded balloon (kg), and mp = the mass of the payload (basket, passengers, and the unexpanded balloon = 265 kg). Assume that the ideal gas law holds (P = PRT), that the balloon is a perfect sphere with a diameter of 17.3 m, and that the heated air inside the envelope is at roughly the same pressure as the outside air. Other necessary parameters are: Normal atmospheric pressure: P = 101,300 Pa, The gas constant for dry air, R = 287 Joules/(kg K), The air inside the balloon is heated to an average temperature, T = 100 C, The normal (ambient) air density, p = 1.2 kg/m². (a) Use a force balance to develop the differential equation for dv/dt as a function of the model's fundamental parameters. (b) Use Euler's method to compute the velocity from t = 0 to 6 s with At = 2 s given the previous parameters along with the initial condition: v(0) = 0. Hint: The buoyancy force is defined by FB = Palg, where V is the volume of the balloon envelope and g = 9.8 m/s2 (standard gravity). (Please solve this problem by hand, not MATLAB