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Question 18.8: A raindrop of mass m starts from rest and falls vertically. ...

A raindrop of mass m starts from rest and falls vertically. When falling with velocity v, it experiences a resistive force of magnitude kv.
i) Express the terminal velocity V_{T} of the raindrop in terms of m, k and g.
ii) Show that, if the raindrop starts from rest, its velocity v after time t is given by

v = V_{T}\left(1  –  e^{-gt/V_{T}} \right).

iii) Assuming the raindrop has a terminal velocity of 5 ms^{-1}, how long does it fall before it has 99% of the terminal velocity?
(Take g as 10 ms^{-2}.)
iv) Derive an expression for the distance fallen in terms of time.

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