A balloon of mass M containing a bag of sand of mass m0 is filled with hot air until it becomes buoyant enough to rise ever so slightly above the ground, where it then hovers in equilibrium. Sand is then released at a constant rate such that all of it is dumped out in a time t0. Find (a) the height of the balloon and (b) its velocity when all the sand has been released. Assume that the upward buoyancy force remains constant and neglect air resistance, (c) Assume that ∈ = m0/M is very small, and find a power series expansion of your solutions for parts (a) and (b) in terms of this ratio, (d) Letting M = 500 kg, m0 = 10 kg, and t0 = 100 s, and keeping only the first-order term in the expansions obtained in part (c), find a numerical value for the height and velocity attained when all the sand has been released.
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