Simple geometric derivations are given for the shape of the “safety domain” boundary for the family of Keplerian orbits of equal energy in a central gravitational field and for projectile trajectories in a uniform field. Examples of practical uses of the envelope of the family of orbits are discussed and illustrated by computer simulations. This material is appropriate for physics teachers and undergraduate students studying classical mechanics and orbital motions.

1.
Italian physicist and mathematician Evangelista Torricelli (1608–1647)
, best known for his invention of the barometer, also studied projectiles extensively. “
Perhaps his most notable achievement in the field of projectiles was to establish for the first time the idea of an envelope: Projectiles sent out at the same point and the same speed in all directions trace out parabolas which are all tangent to a common paraboloid. This envelope became known as the parabola di sicurezza
.”
Philip J.
Robinson
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Evangelista Torricelli
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Planets and Satellites
(
Physics Academic Software, American Institute of Physics
, New York,
1999
). An updated version for the Windows operating system is available as a free download at <http://butikov.faculty.ifmo.ru>.
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