= Escape velocity
{title2=$v_{\mathrm{esc}}=\sqrt{2GM/r}$}
{wiki}
= Escape speed
{synonym}
The <escape velocity> is the least initial <speed> that permits a freely coasting body to move arbitrarily far from an isolated attracting body. Despite its conventional name, it is a <speed>, not a unique <velocity> vector. For a fixed <point mass> $M$, or outside a body with <spherical symmetry>, set <Newtonian gravitational potential energy> zero at infinity. An exterior <test particle> of constant <mass> $m\ll M$ then has
$$
U(r)=-\frac{GMm}{r},\qquad E=\frac12mv^2-\frac{GMm}{r},\qquad v_{\mathrm{esc}}(r)=\sqrt{\frac{2GM}{r}}=\sqrt{\frac{2\mu}{r}}.
$$
Here $\mu=GM$ is the <standard gravitational parameter>, and $r$ is distance from the centre, not height above the surface. Set $r=R$ for surface <escape velocity>. Evaluate <velocity> in the body's rest <inertial frame>. Neglect <drag>, later <rocket thrust>, other bodies' <gravity>, and relativistic effects.<University Physics Volume 1 by William Moebs and others/13.3 Gravitational Potential Energy and Total Energy>[§13.3, equations 13.4–13.6 and “Escape velocity”: infinity reference, small test mass and exterior launch]{ref}
<Conservation of energy> gives the threshold by requiring $E\geq0$: at infinity the <potential energy> tends to zero and the <kinetic energy> cannot be negative. If $E=0$, the outward motion loses <speed> continuously toward zero. If $E>0$, the asymptotic <speed> is $v_\infty=\sqrt{2E/m}=\sqrt{v_0^2-v_{\mathrm{esc}}(r_0)^2}$. If $E<0$, infinity is inaccessible. For a radial outward launch, its outer <turning point> is $r_{\max}=-GMm/E$. For nonradial motion use <angular momentum and the escape energy threshold> to distinguish zero radial <velocity> from zero total <speed>.
Direction still determines the <orbit> and whether the body hits an obstacle. An initially inward <velocity> can cause collision even when $E\geq0$. For an isolated spherical body, an exterior outward launch, or a tangential launch at or above <escape velocity>, has an outward escape branch without intersecting the body. A <parabolic trajectory> is the nonradial threshold path; a <hyperbolic Kepler orbit> has positive <energy>. The formula specifies their required local <speed>, not their orientation.<University Physics Volume 1 by William Moebs and others/13.3 Gravitational Potential Energy and Total Energy>[§13.3, “Energy and gravitationally bound objects”: scalar energy and finite-body collisions]{ref}
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