John Hinch - “Dropping pebbles on the beach”

Problem statement:

• Consider the problem of a projectile (e.g. a pebble) impacting on a granular medium (e.g. sand). Other applications include meteorite impacts,anchors, and artillery fire.

• Open question: Can we explain the experimental observation that the faster the impact, the shorter the time before the projectile stops?

• By writing down a simple model taking into account form drag and solid friction, and doing a Riccati transformation to remove the nonlinear term, we can obtain an expression for the stopping time in terms of
the impact velocity. This takes the form of an integral involving the stopping distance, which is itself the solution of a nonlinear equation involving the impact velocity. In the large impact velocity limit, the integral can be evaluated asymptotically.

• In the general case, we could return to the original equation and do some asymptotics on this directly. In particular, the behaviour of the projectile soon after the impact is easily determined, as is its behaviour just before it comes to rest. Open problem: Do the matching between these two phases.

Suggestions/comments:

• SDH: What are the internal stresses in the granular medium?

• JRO: Comparison with a liquid jet being fired into a granular medium?

People interested in follow-up: JRO, TM, GR