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

Question

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.

Answer 1

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 "m" 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 "m"

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!