# A viscous free fall example¶

## Introduction¶

We consider an object inside a vertical cylinder which contains a viscous fluid. The fluid generates a drag force which limits the speed of the solid and we assume that the force depends linearily on the object speed:

for any where:

• is the speed ,

• is the time ,

• is the maximum time ,

• is the gravitational acceleration ,

• is the mass ,

• is the linear drag coefficient .

The previous differential equation has the exact solution:

for any

where:

• is the altitude above the surface ,

• is the initial altitude ,

• is the initial speed (upward) ,

• is the limit speed :

• is time caracteristic :

The stationnary speed limit at infinite time is equal to :

When there is no drag, i.e. when , the trajectory depends quadratically on :

for any .

Furthermore when the solid touches the ground, we ensure that the altitude remains nonnegative i.e. the final altitude is:

for any .

## Probabilistic model¶

The parameters , , and are probabilistic:

• ,

• ,

• ,

• .

## References¶

• Steven C. Chapra. Applied numerical methods with Matlab for engineers and scientists, Third edition. 2012. Chapter 7, “Optimization”, p.182.

>>> from openturns.usecases import viscous_free_fall as viscous_free_fall