Raindrop Physics: Solving for Terminal Speed of Falling Drop

[S0C0M988]

A raindrop of initial mass Mo starts falling from rest under the influence of gravity. Assume that the drop gains mass at a rate proportional to the product of its instantaneous mass and its instantaneous velocity:

dM/dt = kMV

where k is a constant. Show that the speed of the drop eventually becomes effectively constant, and give an expression for the terminal speed. Neglect air resistance.

 

[OlderDan]

A raindrop of initial mass Mo starts falling from rest under the influence of gravity. Assume that the drop gains mass at a rate proportional to the product of its instantaneous mass and its instantaneous velocity:

dM/dt = kMV

where k is a constant. Show that the speed of the drop eventually becomes effectively constant, and give an expression for the terminal speed. Neglect air resistance.

This is a poorly worded problem. Mass does not suddenly appear out of nowhere. The resulting motion depends on how the drop is acquring the mass. If it is simply condensing other tiny drops that are moving along with it, that is a completely different situation than if it is running into tiny drops that are at rest. I believe the problem is intended to be treated as the raindrop acquiring additional mass from tiny drops that may be considered at rest until bombarded by the larger drop.

Momentum is conserved in every collision. In this problem things are sticking togeter. See what you can do to set up the problem.

 

[S0C0M988]

I have no idea how to even start to set up this problem.

 

[OlderDan]

I have no idea how to even start to set up this problem.

If a raindrop of mass M moving with speed V runs into a bit of water of mass dM, initially at rest, and the two things stick together, how fast will they be moving after the collision assuming no other forces are acting?

It may be that your text is expecting you to assume the validity of the idea that
F = dp/dt = d(MV)/dt with M and V both function of time and do a relatively simple calculus problem. It is a much simpler approach than resorting to momentum conservation fundamentals, but as I stated earlier the assumptions make a difference. You can do the simple calculus problem and hope it is justified, or the more complete problem and justify the result. Choose your approach.