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

1. Feb 19, 2012

### AaronKnight

1. The problem statement, all variables and given/known data
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 = kvm2/3,
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 = kvm2/3".
I am unsure where to begin, any help in pointing me in the right direction would be greatly appreciated.
Thank you.

2. Feb 19, 2012

### 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.

3. Feb 19, 2012

### AaronKnight

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

4. Feb 27, 2012

### 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?

5. Feb 27, 2012

### issacnewton

Well, the density of rainwater is constant.

6. Feb 27, 2012

### dracobook

fair enough thanks.

7. Feb 27, 2012

### issacnewton

post full solution after you are done.