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Problem 2 (Module 1: SLO 1). Figure 1 shows the forces exerted on a hot air balloon system. Formula the drag force as where pa - air density (kg/m2), v- velocity (m/s), A- projected frontal area (m2), and C the dimensionless drag coefficient (0.47 for a sphere). Note also that the total mass of the balloon consists of two components: where mo 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 temperarture, T 100°C. The normal (ambient) air density, p 1.2 kg/m3. (a) Use a force balance to develop the differential equation for du/dt as a function of the models fundamental parameters. (b) At steady-state, calculate the particles terminal velocity. Figure 1: Forces on a hot air balloon: Fs buoyancy, Fa weight of the hot air (gas), F, = weight of the payload (including the balloon envelope), and F-drag. Note that the direction of the drag always opposes the direction of motion of the balloon.

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