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blalien

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**[SOLVED] Falling raindrop**

## Homework Statement

This problem is from Gregory. Yeah, classical mechanics is not being kind to me this week. I swear I'm not coming here for every single problem.

A raindrop falls vertically through stationary mist, collecting mass as it falls. The raindrop remains spherical and the rate of mass accretion is proportional to its speed and the square of its radius. Show that, if the drop starts from rest with a negligible radius, then it has constant acceleration g/7.

m: the mass of the raindrop

r: the radius of the raindrop

v: the velocity of the raindrop

## Homework Equations

Because the raindrop is a sphere: [tex]m = \rho 4/3 \pi r^3[/tex] for some constant [tex]\rho[/tex]

[tex]m'(t) = k v r^2[/tex] for some constant [tex]k[/tex]

## The Attempt at a Solution

Plugging in our equation for m, we have the differential equation

[tex]m' = k v (\frac{3m}{4\pi \rho})^{2/3}[/tex]

With initial conditions m = 0 (or close enough) and v = 0.

We need another differential equation, since both m and v are nonconstant.

Thinking about this intuitively, I don't see why the raindrop isn't just falling with acceleration g. I mean, they're just particles falling, with no other external force. But since I apparently can't assume that the acceleration is g, I see no logical reason to assume that the acceleration is even constant.

So I guess my question is, how can I find the second differential equation?

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