# 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:

• ,

• ,

• ,

• .

We can load this model from the use cases module as follows :

>>> from openturns.usecases import viscous_free_fall
>>> # Load the viscous free fall model
>>> fm = viscous_free_fall.ViscousFreeFall()


## Examples based on this use case¶   