Question

A hot air balloon of total mass M (including passengers and luggage) is moving with a downward acceleration of magnitude a. As it approaches a mountain, the captain needs to accelerate upwards. He decides to throw enough ballast over board to achieve an upward acceleration of magnitude a/2. What fraction of the initial mass does he have to drop?

Answer 1

When balloon moves in the downward direction two forces acts on it.

i) Force exerted by air in the upward direction

ii) Weight

According to newton’s second law of motion:

Sum of forces = Ma

W – F = Ma

Mg – F = Ma …….. (i)

when some of the mass m is dropped and balloon is moving in upward direction with acceleration a/2 then,

F – W = (M-m)a/2

F – (M-m)g = (M-m)a/2

F – Mg + mg = Ma/2 – ma/2 ….. (ii)

Adding equation (i) and (ii)

mg = M(3a/2) – ma/2

m(g + a/2) = M(3a/2)

m = M(3a/2)/(g + a/2)