# Halley’s minimal kinetic energy problem for projectile motion with drag quadratic in speed

• Robert Kantrowitz
• Michael M. Neumann
Article

## Abstract

In papers published in the Philosophical Transactions in the late 1600s, Edmond Halley highlighted the problem of delivering a projectile to its intended target with minimal kinetic energy upon impact. Halley accounted for the force of gravity acting on the projectile, but ignored the retarding effect of air resistance. In this article, we revive the optimization problem raised by Halley, here allowing for drag that is quadratic in speed. It turns out that there are remarkable parallels between the optimal flight curve in the case of no air resistance and that in which air resistance is quadratic in speed.

## Keywords

Projectile ballistics Linear drag Quadratic drag Halley’s minimal kinetic energy problem

## Mathematics Subject Classification

Primary 26A06 Secondary 49K15

## Notes

### Acknowledgements

We thank the referees for their careful reading and valuable suggestions.

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