Vertical Launch of a Grappling Hook

Abstract

Problems involving chains, cables, or ropes that are dropped, folded, or pass around pulleys attract ongoing interest, in part because they can become variable-mass situations if the chain is partitioned into sections for analysis. Less attention has been paid to trying to intentionally project the end of a string as far as possible. Here we consider the launch of a grappling hook connected to a long flexible rope, such as the cartoon character Batman might do. As a real-life example of their use, rocket-propelled ropes with grapnels were employed to ascend French cliffs on D-Day, although the wetness of the lines made the task more difficult than expected. Line launchers have also been used to access the roofs of burning buildings.

Unfurling a rolled-up sheet from a dispenser at constant speed

“…The analysis could also be generalized to account for kinetic friction between the inner tube and the rod or bracket on which it is mounted [7], or for rolling friction in bearings [8] used to couple the roll to the axle. The problem shares some features with those involving yo-yos [9], spools pulled by their threads so that they roll without slipping along a table [10,11], carpet rolls given a push so that they unfurl along a floor [12,13], harpoon guns [14], or yarn being pulled off the spindle of a spinning wheel. One could also imagine a rolling wheel ejecting layers of material (such as mud or truck-tire treads) from the top of the wheel.…”