Alokananda's Sand Tower

Classical Mechanics Level 5

Directly below her \(10\text{ m}\) tall tower, Alokananda has piled up a heap of sand \(1\text{ m}\) high.

When she drops a ball of density \(\rho\) from the top of the tower, the ball does reach the bottom of the sand (whose density, by the way, is \(\sigma\)) pit, but by then, it is completely exhausted of its kinetic energy.

Modelling the heap of sand as a pool of fluid and assuming that the Archimedes' principle is the only reason for the resistance provided by the sand, what is the value of \(\dfrac{\sigma}{\rho}\)?

Assume that the ball has a positive radius negligible compared to the height of fall. Further, also assume that the ball immerses completely and no energy is lost in the air-sand transition.

Inspired by Alokananda Sarkar.

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