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?

Question

Rory

Physics

1 February, 07:33

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?

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Aspen Landry · 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)