Mechanics limiting problem with variable radius.

Okay so I'm trying to solve this problem and have ran into some difficulties.



Using impulse change of momentum principles I managed to figure out that the equation of motion for the hailstone is

$\displaystyle \frac{dv}{dt}+\frac{v}{m}\frac{dm}{dt}=g$ however I don't know how to work out $\displaystyle \frac{dm}{dt}$ I thought it must have something to do with the differential equation given about the radius in the question but I couldn't figure it out and got stuck.

Presumably when I figure out $\displaystyle \frac{dm}{dt}$ I can just solve to find $\displaystyle v(t)$ and take the limit as $\displaystyle t \rightarrow \infty$ and the answer should drop out.

Any help?

Okay so I'm trying to solve this problem and have ran into some difficulties.



Using impulse change of momentum principles I managed to figure out that the equation of motion for the hailstone is

$\displaystyle \frac{dv}{dt}+\frac{v}{m}\frac{dm}{dt}=g$ however I don't know how to work out $\displaystyle \frac{dm}{dt}$ I thought it must have something to do with the differential equation given about the radius in the question but I couldn't figure it out and got stuck.

Presumably when I figure out $\displaystyle \frac{dm}{dt}$ I can just solve to find $\displaystyle v(t)$ and take the limit as $\displaystyle t \rightarrow \infty$ and the answer should drop out.

Any help?

Use the fact that m=k*(4/3 pi r^3)

Where does the k come from? (4/3 pi r^3) is obviously the volume? so is k the density? is this obvious or am I just going mad???

Mass = density x volume

but note that in your DE you have (v/m) multiplying dm/dt so even though you don't know the density, it will cancel out in your working - just call it $\rho$ or something.

You can work out rate of change of volume using the chain rule.