In classical mechanics and ballistics, the parabola of safety or safety parabola is the envelope of the parabolic trajectories of projectiles shot from a certain point with a given speed at different angles to horizon in a fixed vertical plane. The fact that this envelope is a parabola had been first established by Evangelista Torricelli and was later reproven by Johann Bernoulli using the infinitesimal calculus methods of Leibniz.

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The paraboloid of revolution obtained by rotating the safety parabola around the vertical axis is the boundary of the safety zone, consisting of all points that cannot be hit by a projectile shot from the given point with the given speed.

In 2D and shooting on a horizontal plane, parabola of safety can be represented by the equation

${\displaystyle y={\frac {u^{2}}{2g}}-{\frac {gx^{2}}{2u^{2}}}}$

where ${\displaystyle u}$ is the initial speed of projectile and ${\displaystyle g}$ is the gravitational field.

• Focus of the parabola is the shooting position.
• Maximum height (${\displaystyle H}$) can be calculated by absolute value of ${\displaystyle c}$ in standard form of parabola. It is given as ${\displaystyle H=|c|={\frac {u^{2}}{2g}}}$
• Range (${\displaystyle R}$) of the projectile can be calculated by the value of latus rectum of the parabola given shooting to the same level. It is given as ${\displaystyle R=|4c|={\frac {u^{2}}{g}}}$

• Philip Robinson, On the Geometrical Approach to Projectile Motion, The Mathematical Gazette, Vol. 82, No. 493, 1998, pp. 118–122

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