# How do you determine how fast an item sinks in water ?

1. Jul 10, 2008

### bertyboy

Could someone show me the simple formula, showing how quickly
an item sinks in water.
Presumably the formula incorporates some sort of drag co-efficient,
which relates to the shape of the body ?

2. Jul 10, 2008

### arildno

Check up on Stokes' law of resistance.

3. Jul 10, 2008

### DeepSeeded

It is proportional to the shape and denisty of the object. For a sphere it is proprotional to Diameter * Diameter * Pi (The cross sectional area of the center of the sphere) and the density of course.

Check out lecture 27 and 28 for the exact equations here: http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/index.htm [Broken]

Last edited by a moderator: May 3, 2017
4. Jul 10, 2008

### stewartcs

The settling velocity can vary depending on the size of the object. As arildno pointed out, Stoke's Law is a good place to start. However, if the size of the object is very small, it may be influenced by Brownian motion.

Intermediate Law is another approach.

CS

5. Jul 17, 2008

### bemortu

Hi.

You can start from the following equation for the drag force
$$F_D=C_DA \frac{\rho v^2}{2}$$

Here $$v$$ is the speed og the object and $$\rho$$ is the density of the water. $$A$$ is the referense area used to define the drag coefficient $$C_D$$, usually the projected cross-section area in the direction of the velocity.

Combine this with Newtons second law:
$$ma=mg-F_D$$
Then you have an expression for the acceleration of the object. If you want the final velocity you take $$a=0$$ and solve for $$v$$.

The tricky part here is that the drag coefficient $$C_D$$, in general, depends on $$v$$ (or actually on the Reynolds number). If Re<1 you can use Stokes law, but that would require a quite small (or very light) object. For more typical engineering type appications a constant $$C_D$$ is often applicable. For a sphere $$C_D\approx0.44$$ for a quite large range of Reynolds numbers.

If you don't know at all what Reynolds number you expect you may have to guess what relation to use for drag coefficient, calculate the velocity, calculate the Reynolds number and then check if the drag law you used is compatible with the this Reynolds number. If not, try another drag law.