The change of mass with respect to time of a rain drop falling though a cloud

[AaronKnight]

Homework Statement

A raindrop is falling through a cloud, collecting water as it falls. If the raindrop has mass
m and velocity v, show that
mg = vdm/dt +mdv/dt.
Assume that the drop remains spherical and that the rate of accretion is proportional to the
cross-sectional area of the drop multiplied by the speed of fall. Show that this implies that
dm/dt = kvm^2/3,
for some constant k. Using this equation and the assumption that the drop starts from rest
when it is infinitesimally small, find m as a function of x. Hence show that the acceleration
of the raindrop is constant and equal to g/7.


I have managed to show that mg = vdm/dt +mdv/dt easily, but I am at a lose for the second part where the question asks "Show that this implies that dm/dt = kvm^2/3".
I am unsure where to begin, any help in pointing me in the right direction would be greatly appreciated.
Thank you.

 

[issacnewton]

hi aron, the rate of accretion is the rate at which mass is being put on the drop. so the rate of accretion is $\frac{dm}{dt}$. The area of cross section of the sphere is $\pi r^2$ , where r is the radius. So we are given that

$$\frac{dm}{dt} = k_1 v\; \pi r^2$$

where k₁ is just the constant of proportionality. now since the density of the drop remains constant, use that fact to write r² in terms of density and the mass of the drop. whatever constants you encounter, just merge it and call it k.

 

[AaronKnight]

Thank you, it was the last step of writing r² in terms of the mass and density that I was missing out.

 

[dracobook]

Hi. Sorry I know this thread has already been answered. Just one quick question: How do you know that the density of the drop remains constant?

 

[issacnewton]

Well, the density of rainwater is constant.

 

[dracobook]

fair enough thanks.

 

[issacnewton]

post full solution after you are done.