Mechanics question using a differential equation

Question

I have come across a question on a first-year undergraduate mechanics paper:

I am unsure of how I should approach this. How can I use the radius to create a link between this and the speed of the hailstone? Especially since air resistance is being neglected, why is the radius still important in the question since no matter how big the radius is, the surface area doesn't create any further drag?

My initial thoughts would be to calculate a time using the differential equation except I am unsure what to consider in order to find such a value. Any help would be appreciated!

Answer 1

The mass of the spherical hailstone is $$m=\frac 43\pi r^3\rho$$ where$\rho$ is the density. So $$\frac{dm}{dt}=4\pi r^2\rho \frac{dr}{dt}=3mk$$

Meanwhile the equation of motion is $$\frac{d}{dt}(mv)=mg$$ $$\implies m\frac{dv}{dt}+v\frac{dm}{dt}=mg$$

For the limiting speed we require $\frac{dv}{dt}\rightarrow0$

In which case, $$v(3mk)\rightarrow mg$$

So the terminal velocity is $\frac{g}{3k}$

Answer 2

Keep in mind that the equation you are solving is:

$$f=mg=\frac{dp}{dt}=\frac{d}{dt}(mv)$$ with $$m=m(t)$$