Falling Water Droplet Homework: Show Radius & Speed Increase Linearly over Time

[Varnson]

Homework Statement

Assume that a water droplet falling through a humid atmosphere gathers up mass at a rate that is proportional to its cross-sectional area A. Assumethat the droplet starts from rest and that its initial radius R0 is so small that it suffers no resistive force. Show that its radius adn its speed increase lineraly with time.

Homework Equations

A= pi[(4/3)pi]^(-2/3)*Vol^(2/3)

The Attempt at a Solution

I found the above equation to relate the area to the volume. I am stuck on this and do not know which direction I should head in.

 

[Gokul43201]

That's a poor question; it's forcing you to make a plainly unphysical assumption, namely that $\dot{m}(t)=km^{2/3}(t)$, when a much better assumption would be $\dot{m}(t)=kv(t)m^{2/3}(t)$. However, the latter does not produce an easy answer, while the former does.

That said ... the above differential equation is trivially solved to give you the first answer that you need. For the second answer, you need to write an equation for momentum conservation and plug in the result from the first answer.