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 FD 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 FD = –k z, independent of velocity. We thus find the following equation of motion
where mA 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+mA)]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 mA is zero within measurement precision.
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