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Problem 49 Easy Difficulty

When a raindrop falls, it increase in size and so its mass at time $ t $ is a function of $ t, $ namely, $ m(t). $ The rate of growth of the mass is $ km(t) $ for some positive constant $ k. $ When we apply Newton's Law of Motion to the raindrop, we get $ (mv)' = gm, $ where $ v $ is the velocity of the raindrop (directed downward) and $ g $ is the acceleration due to gravity. The terminal velocity of the raindrop is lim $ _{t \to \infty} v(t). $ Find an expression for the terminal velocity in terms of $ g $ and $ k. $


$$\lim _{t \rightarrow \infty} v(t)=g / k$$


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