A hot-air balloon of mass M is descending vertically with downward acceleration a...?

How much mass must be thrown out to give the balloon an upward acceleration a (same magnitude but opposite direction)? Assume that the upward force from the air (the lift) does not change because of the mass (the ballast) that is lost.A hot-air balloon of mass M is descending vertically with downward acceleration a...?
Free body diagram, balloon going down (-):

Sum of forces: L - W

F = m * a

L - W = M * (-a); W = M * g

solve for Lift, L

L = M * (g - a)



Free body diagram, balloon going up (+):

Sum of forces: L - W

F = m * a

Let %26quot;m%26quot; be the new mass of the balloon system

L - W = m * a; W = m * g

solve for Lift, L

L = m * (g + a)



Lift is constant, so set both L's equal:



M * (g - a) = m * (g + a); solve for new mass %26quot;m%26quot;



m = [ (g - a) / (g + a) ] * M; the problem asks for the mass/ballast thrown out, so this is just



ballast = M - m = M - [ (g - a) / (g + a) ] * M, or



ballast = {1 - [ (g - a) / (g + a) ] } * M



Good Luck!

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