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

  • Robert KantrowitzEmail author
  • Michael M. Neumann


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.


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

Mathematics Subject Classification

Primary 26A06 Secondary 49K15 



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


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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHamilton CollegeClintonUSA
  2. 2.Department of Mathematics and StatisticsMississippi State UniversityMississippi StateUSA

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