**Drag in Dry
Quicksand**

We developed a simple force model to describe the dynamics of
the ball in the sand and the parameter dependence of the final depth. The ball
is accelerated by gravity and at the same time experiences a drag force *F*_{D} from the sand grains. For
simplicity, we assume a Coulomb drag due to the normal forces from the side,
which linearly increases with the depth *z*.
We therefore have *F*_{D} = –k *z*,
independent of velocity. We thus find the following equation of motion

,

where *m*_{A} is the added mass. This equation has to be
supplemented by the boundary conditions *z*(0)
= 0 and = 0.

Integration
immediately gives the final depth

.

i.e., a linear dependence of the
depth on the mass. This is in agreement with the experimental findings (see top figure),
from which we can read off k = (13.3 ± 0.5) N/m. The full solution of the equation of motion is:

for 0 ≤ *t* ≤ p/w with w = [k/(*m*+*m*_{A})]^{1/2}. This solution describes the dynamics
extremely well, as is seen from the bottom figure. From the fitted w we obtain that the added mass *m*_{A} is zero within measurement precision.

*Homepage:*

*See also the web pages of the research
group:*

Physics of Fluids – Department of Science and Technology – University of Twente – The