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14 June, 01:02

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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  1. 14 June, 01:22
    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)
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