Math Is Fun Forum

Reuel

Member

Registered: 2010-11-28

Posts: 178

Terminal Velocity of a Raindrop

This is a problem from an introductory ODE course.

"When a raindrop falls, it increases in size and so its mass at time t is a function of t, namely m(t). The rate of growth of the mass is km(t) for some positive constant k. When we apply Newton's Law of Motion to the raindrop, we get (mv)' = gm, where v is the velocity of the raindrop (directed downward) and g is the acceleration due to gravity. The terminal velocity of the raindrop is the limit as t goes to infinity for v(t). Find an expression for the terminal velocity in terms of g and k."

For those in the wake, the full solution is shown in the posts below.

Last edited by Reuel (2011-01-25 13:34:51)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Terminal Velocity of a Raindrop

Hi Reule;

This is what I would do.

Solved it using Mathematica.

Can solve for the constant c by substituting v(0)=0.

1) becomes:

The limit of v(t) as t approaches infinity is g / k

So the terminal velocity is g / k.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Terminal Velocity of a Raindrop

bobbym,

Thanks! I actually thought your first edit, the first three lines, was your final response and I worked out most of the rest myself. Which is good for me. But I do not know how to use Mathematica. I use Maple. Do you know anything about solving equations with generic constants such as g and c and k in Maple?

Thanks a lot for your help! I am glad I am now understanding this. I have not taken physics so all the stuff about gravity and proportioned mass to time kind of threw me off.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Terminal Velocity of a Raindrop

Yes, please post the exact problem you want to solve with it and what you did and the result.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Terminal Velocity of a Raindrop

The one you just solved in Mathematica is a good example.

In Maple I typed

to open the DETool box, then

In hopes of solving the differential equation. But I haven't really learned much about Maple's DE tools yet.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Terminal Velocity of a Raindrop

What version of Maple do you have?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Terminal Velocity of a Raindrop

13.

Sorry to keep withholding information.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Terminal Velocity of a Raindrop

Here is how you do it on the worksheet not the classical the better one that accepts pretty print input.

This is the general solution:

This is the particular solution

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Terminal Velocity of a Raindrop

Neat! Now would this have been an easily-found solution by hand or no? I am still learning separation of variables and all that to do with natural logs, general solutions, and specific solutions.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Terminal Velocity of a Raindrop

Remember to use diff(v(t),t) to enter it.

As for by hand, I have not done one of these in years and years. You cannot separate the variables but since this is a linear first order DE it should respond to a book method.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Terminal Velocity of a Raindrop

Thanks again for your help! Now I can see how it all goes together.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Terminal Velocity of a Raindrop

Hope it is okay, see you later.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: Terminal Velocity of a Raindrop

P. S. The solution of the ODE can be solved by hand. Here is how:

And solving for v(t) gives the general solution

Letting v(0) = 0 finds the particular solution

And you can factor out the g/k:

And so, in stating the answer as the original limit in terms of g and k,

In words: "The terminal velocity of a raindrop with mass m and velocity at time t is directly proportional to the acceleration due to gravity and inversely proportional to some constant k."

Last edited by Reuel (2011-01-25 05:48:10)