Potential energy ($\mathbf{F}=-\nabla V$)

Potential energy assigns energy to a system's arrangement, so a change of arrangement accounts for the work done by a conservative force. For a time-independent potential energy $V(\mathbf{r})$,The gradient points toward increasing potential energy; the force points toward decreasing potential energy. Only differences matter: adding a constant to $V$ changes neither the force nor its work. This endpoint rule follows by substituting $-\nabla V$ into the line integral for work. ^[ref]

Near Earth's surface, stipulate constant gravitational acceleration $g=9.81m/{\mathrm{s}}^2$ and measure height $h$ upward from a chosen zero. Then gravitational potential energy is $V=mgh$. Raising a $2\mathrm{kg}$ mass by $0.5\mathrm{m}$ increases its gravitational potential energy by $9.81\mathrm{J}$, while gravity does $-9.81\mathrm{J}$ of work. These are computed model values, not measurements of a particular lift. An ideal spring obeying Hooke's law instead has $V=k{x}^2/2$, with displacement $x$ from its relaxed state and spring constant $k$: $k=100N/m$ and $x=0.2\mathrm{m}$ give $V=2\mathrm{J}$ relative to $x=0$. ^[ref]

In uniform gravity a body's gravitational potential energy is its mass times gravitational acceleration times the Cartesian height of its centre of mass. If that height is represented by an axis distance $h$ and tilt angle $\theta$, it is $h\cos \theta$ and the energy becomes $Mgh\cos \theta$. Treating $h$ simultaneously as a Cartesian height would introduce an extra, unjustified cosine.

Named after John Lennard-Jones, this is a pair potential energy combining steep short-range repulsion with nonretarded London dispersion force attraction. Its minimum is $U=-ϵ$ at $r=2^{1/6}\sigma$, found by differentiation. The power twelve is a convenient model for repulsion, rather than an exact consequence of the Pauli exclusion principle. The power six in the attractive energy follows from second-order coupling of atomic electric dipoles, before retardation matters.

A quadratic potential energy about a stable equilibrium. In one dimension $k>0$ is the restoring stiffness and the force is $-dV/dx=-kx$. A mass $m$ undergoing undamped motion has angular frequency $\sqrt{k/m}$. A smooth potential with a nondegenerate local minimum is approximately harmonic for sufficiently small displacements; higher-order terms describe departures from simple harmonic motion.

A region of lower potential energy surrounded by higher potential energy; a particle must gain sufficient energy to escape classically.

A potential energy landscape with two local minima separated by a barrier. In a symmetric one-dimensional case, its two lowest wavefunctions are an even function and an odd function, with their energies split by quantum tunnelling. A two-mode approximation for a Bose gas retains these two orbitals when higher energy levels are sufficiently remote.

A confining Quantum Hamiltonian on $x>0$ with $a>0$, potential $a\sqrt{x}$ and a Dirichlet boundary condition at zero has natural length $\ell =({\hslash }^2/(2ma){)}^{2/5}$ and energy $E_{\ast }=({\hslash }^2{a}^4/(2m){)}^{1/5}$. Rescaling $x=\ell y$ gives $H/E_{\ast }=-d^2/d{y}^2+\sqrt{y}$, so every energy eigenvalue scales as $E_{\ast }$. The potential derivative diverges at the wall even though the potential itself is finite there.

This potential energy generates a repulsive central force $F_r=-V^{\prime }(r)=2c/r^3$. It is an inverse-square potential but an inverse-cube force; the powers must not be confused with the Coulomb or gravitational potential proportional to $1/r$.

For $V(r)=c/r^2$ with $c>0$, nonzero physical angular momentum $h$ and positive energy $E$, put $\beta =\sqrt{1+2mc/h^2}>1$ and $r_{\min }=\sqrt{(c+h^2/(2m))/E}$. Direct integration of the effective potential radial equation gives the displayed orbit. It has one radial minimum and two straight-line asymptotes separated by polar angle $\pi /\beta$. The repulsive scattering deflection is $\pi -\pi /\beta$; no bounded or circular orbit exists at finite positive energy.

The potential energy associated with gravitational interactions. For a self-gravitating spherical body with a fixed dimensionless density profile, $E_{\mathrm{grav}}=-CG{M}^2/R$ for a positive profile-dependent constant $C$; a uniform sphere has $C=3/5$. The $R^{-1}$ scaling competes with the same radius dependence of ultrarelativistic electron degeneracy pressure in a white dwarf, giving the Chandrasekhar mass scale.

The portion of a fluid system's potential energy that can be released by admissible rearrangement relative to a stable reference state. In a uniformly stratified incompressible fluid, the quadratic density is $A={\rho }_0{N}^2{\xi }_z^2/2={\rho }_0{b}^2/(2N^2)$, where ${\xi }_z$ is vertical fluid displacement and $b$ the buoyancy perturbation. It is positive for stable stratification and measures the restoring energy of internal gravity waves.

The background potential energy is the gravitational potential energy of the statically stable state obtained by adiabatically sorting a fluid's parcels. The difference $A=P-P_b$ is available potential energy. Pure rearrangement changes $A$ but not $P_b$; irreversible density mixing can increase $P_b$. Molecular exchange with internal energy must be separated when defining an irreversible mixing rate.

A potential function $V$ assigns a scalar to each point of a configuration or spatial domain. In mechanics it represents potential energy, and its negative gradient gives the corresponding conservative force.

A central potential depends only on distance from a fixed centre, $V(\mathbf{x})=V(|\mathbf{x}|)$. Its force is radial, and rotational symmetry implies conservation of angular momentum.

Adding an inverse-cube force can preserve radial motion while rescaling angular motion. For potential $-A/r+B/r^2$ and angular momentum $L$, the radial effective potential equals that of $-A/r$ with effective angular momentum $\sqrt{L^2+2mB}$ when this is real.

A force is conservative when its work depends only on the endpoints. On a simply connected domain this is equivalent to $F=-\nabla V$ for a potential energy $V$.

A nonconservative force has path-dependent work. On a simply connected domain, a continuously differentiable force is nonconservative exactly when its curl does not vanish identically.

Here inverse-square potential means the potential energy associated with an attractive inverse-square law for a central force: the potential energy itself varies as inverse distance. For $k>0$ and position $\mathbf{r}$ with $r=|\mathbf{r}|>0$,The gradient makes the distinction explicit: doubling $r$ halves the magnitude of potential energy but quarters the magnitude of force. The neighboring repulsive inverse-square potential instead has potential energy proportional to $r^{-2}$ and an inverse-cube force; the two naming conventions must be distinguished.

For Newton's law of universal gravitation, $k=GMm$, where $G$ is the gravitational constant. Treat the attracting mass $M$ as an isolated fixed point mass, or as a body with spherical symmetry and exterior test particle of constant mass $m\ll M$. The spherical shell theorem gives the same exterior Newtonian gravitational potential energy $U=-GMm/r$ without requiring uniform mass density. Here $r$ is measured from the centre, and the formula applies outside the body; it does not describe its interior. The Newtonian gravitational potential per unit mass is $\Phi =U/m=-GM/r$.^[ref]

The negative sign records binding relative to infinite separation, rather than a negative kinetic energy. Moving from $r_1$ to $r_2>r_1$ increases potential energy byBecause the force is a conservative force, this work depends on the endpoints, including for a curved path. Integrating $dV/dr=k/r^2$ with zero at infinity gives $V=-k/r$; adding a constant leaves the force unchanged but shifts the numerical escape energy threshold by that same constant. This is why the reference at infinity must accompany an escape velocity calculation.

For an exact synthetic comparison, stipulate the standard gravitational parameter $\mu =GM=8{\mathrm{m}}^3/{\mathrm{s}}^2$ and a $1\mathrm{kg}$ test particle. At $r=2\mathrm{m}$, $U=-4\mathrm{J}$ and the inward force has magnitude $2\mathrm{N}$; at $r=4\mathrm{m}$, $U=-2\mathrm{J}$ and the inward force has magnitude $0.5\mathrm{N}$. The increase is $2\mathrm{J}$ of potential energy. These stipulated parameters are an arithmetic example, not observations of a particular body. The escape-energy balance diagram uses the same parameters.

In the isolated model, conservation of energy and angular momentum conservation reduce the exterior test particle's motion toHere $\epsilon$ is specific orbital energy and $h$ is specific angular momentum. The effective potential is ${\Phi }_{\mathrm{eff}}=h^2/(2r^2)-\mu /r$. A zero radial velocity is a radial turning point, and need not mean zero total speed: at the periapsis or apoapsis of a nonradial Kepler orbit, tangential velocity remains. Thus the vertical gap between $\epsilon$ and $-\mu /r$ represents total kinetic energy per unit mass, whereas the gap above ${\Phi }_{\mathrm{eff}}$ represents its radial part.

For $h>0$, the orbital eccentricity obeys $e^2=1+2\epsilon h^2/{\mu }^2$. Negative $\epsilon$ gives an elliptic orbit, including a circular orbit at its minimum allowed energy; zero $\epsilon$ gives a parabolic trajectory; positive $\epsilon$ gives a hyperbolic Kepler orbit. The effective potential tends to zero from below at infinity, so its outer region creates no extra positive escape barrier. This explains why escape velocity depends on energy even though angular momentum changes the path. The $h=0$ case is radial motion, rather than a nondegenerate parabola or hyperbola; inward radial motion can end in collision.

One motivating laboratory demonstration compares two balls initially at the same height. A switch releases one vertically when the other leaves a horizontal track; arranging their paths to meet gives a collision as the readout of equal downward displacement over equal time. Equal initial height and synchronized release control the comparison. This tests the local independence of horizontal motion and free fall, under nearly constant gravitational acceleration; it does not measure an inverse-square law.^[ref]

A complementary astronomical comparison uses the Moon's distance and orbital period to infer inward centripetal acceleration, $a=4{\pi }^{2}r/T^2$, and compares it with surface gravitational acceleration scaled by $(R/r{)}^2$. The approximately sixtyfold distance ratio gives the approximately $1/3600$ prediction. The inference links terrestrial free fall to a lunar orbit. Distance calibration matters; the circular orbit approximation and the Sun's tidal force limit this elementary comparison.^[ref]

Lunar laser ranging gives a more discriminating orbit readout. An observatory directs short laser pulses at a retroreflector array on the Moon, and a detector records returning photons. Round-trip time, statistically combined over several returns, constrains distance; pressure and temperature accompany the readout. The fitted model includes Newtonian gravity, terrestrial and lunar rotation, geophysical effects, and corrections from general relativity. Comparing orbit determination residuals across stations and model choices checks consistency. The inspected study reports $1.8\mathrm{cm}$ weighted root mean square residuals for its 2005–2011 ranges, rather than for a bare isolated inverse-square potential. It tests the equivalence principle using the Earth and Moon's response to the Sun; it is not a direct escape velocity measurement. Uneven coverage near new and full Moon, radiation pressure, and thermal expansion of reflector mounts limit that inference.^[ref]

A retroreflector returns incident light approximately toward its source. Corner-cube arrays make the lunar laser ranging round trip possible.

Three mutually perpendicular mirrors reverse a ray's direction when it encounters all three. With orthonormal normals $a,b,c$, write the reflections as $I-2P_a$, $I-2P_b$, $I-2P_c$. Their orthogonal projection matrices have pairwise products zero and sum to $I$ by the resolution of the identity in an orthonormal basis. The product of the three reflections is therefore $-I$. It reverses every incident direction independently of the order in which the mirrors are met. Real apertures restrict which incoming rays actually reach all three reflecting faces. Such retroreflectors provide the returning-light geometry in lunar laser ranging.

The even potentialhas a cusp at the origin and approaches the finite value $V_{\infty }$. Motions below that value oscillate between symmetric turning points; motions above it escape.

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 hasHere $\mu =GM$ 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.^[ref]

Conservation of energy gives the threshold by requiring $E\ge 0$: 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 $E\ge 0$. 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.^[ref]

For a spherical nonrotating asteroid of the same mean mass density as a reference planet of radius R and surface gravity g, the mass scales as r cubed. Its surface escape speed therefore obeys $v_{\mathrm{esc}}^2=2g{r}^2/R$. A take-off speed $v_J$ permits escape for $r\le v_J\sqrt{R/(2g)}$. If that same take-off speed gives ballistic height H above the reference planet, $v_J^2=2gH$, and the limiting asteroid radius is $\sqrt{RH}$. A chosen H is an explicit performance estimate, not a universal human constant.

Equating the molecular root-mean-square speed to the surface escape velocity gives $3k_{B}T/m=2GM/R$, or $T=2mGM/(3k_{B}R)$. This compares speed scales; the high-speed tail can escape below that temperature, and molecular dissociation can invalidate the assumed species at high temperature.

Use the synthetic standard gravitational parameter $\mu =8{\mathrm{m}}^3/{\mathrm{s}}^2$ and launch distance $r_0=2\mathrm{m}$. Then escape speed is $2\sqrt{2}m/s$. Three outward radial launches give exact comparisons: $v_0=2m/s$ has specific orbital energy $-2J/kg$ and stops outward at $r=4\mathrm{m}$; $v_0=2\sqrt{2}m/s$ has zero specific orbital energy; $v_0=2\sqrt{3}m/s$ has specific orbital energy $+2J/kg$ and asymptotic speed $2m/s$. The second case has speed $2m/s$ at $r=4\mathrm{m}$. Doubling $r_0$ divides escape speed by $\sqrt{2}$, whereas doubling $M$ at fixed $r_0$ multiplies it by $\sqrt{2}$. Changing the test particle's mass changes $E$ but cancels from the escape speed.

The same initial distance can lead to strikingly different orbits. Let $s=v_0/\sqrt{\mu /r_0}\ge 1$ and launch tangentially in one orbital plane. Then periapsis is the launch point, specific angular momentum is $h=r_0{v}_0$, andwhere $f$ is true anomaly. These follow by combining the initial specific orbital energy with the orbital eccentricity relation and the Kepler orbit equation. With $\mu =8{\mathrm{m}}^3/{\mathrm{s}}^2$ and $r_0=2\mathrm{m}$, the initial speeds $2$, $\sqrt{6}$, $2\sqrt{2}$, and $2\sqrt{3}m/s$ give $e=0,1/2,1,2$: a circular orbit, an elliptic orbit with apoapsis distance $6\mathrm{m}$, a parabolic trajectory, and a hyperbolic Kepler orbit. The two open paths have asymptotic speeds zero and $2m/s$ respectively. Their different directions at large distance are part of the Kepler orbit, not an extra scalar escape threshold.

“Arrives at infinity with zero speed” is an asymptotic statement. For outward radial motion at zero specific orbital energy,Every finite distance takes finite time, but $t(r)\to \infty$ as $r\to \infty$. There is no instant of rest at an infinitely distant destination. Differentiating the displayed solution gives $\ddot{r}=-\mu /r^2$, independently checking Newton's second law. In the synthetic example, reaching $r=4\mathrm{m}$ takes exactly $(4-\sqrt{2})/3\mathrm{s}\approx 0.861929\mathrm{s}$. For nonzero specific angular momentum, the Barker equation gives the corresponding infinite-time limit along a parabolic trajectory. Positive-energy escape also takes infinite time to reach infinity; it retains nonzero asymptotic speed.

For a circular orbit, gravity supplies centripetal acceleration, giving $v_c^2/r=\mu /r^2$. ThusA body in a circular orbit is bound despite its continuous free fall. At the same $r$, escape needs twice its kinetic energy per unit mass. An instantaneous forward tangential velocity increment from the circular orbit requires $\Delta v=(\sqrt{2}-1)v_c$, rather than the full escape speed from rest. In the synthetic example this is exactly $2(\sqrt{2}-1)m/s\approx 0.828427m/s$.^[ref]

Escape speed is a coasting threshold, not a required continuously maintained speed or a measure of rocket propellant. A powered vehicle can start at rest and acquire energy gradually. The rocket equation relates an ideal velocity increment to rocket exhaust speed and rocket mass ratio; rocket thrust, drag, and the timing and direction of propulsion affect a real ascent. Rotation also changes the relation between surface-relative velocity and the inertial frame used in the calculation.

Reaching another body is a finite-destination spacecraft trajectory problem. In the synthetic radial example, a launch at $2m/s$ reaches $r=4\mathrm{m}$ while remaining bound. A spacecraft can similarly encounter the Moon on a bound geocentric transfer; actual planning must include the Moon's gravity, target motion, arrival velocity, and any propulsion needed for capture. Escaping Earth alone does not imply escaping the Sun, because the relevant reference frame and attracting mass have changed. The isolated formula describes one gravitational well; a full mission requires more than its scalar threshold.

Operational evidence comes from reconstructing a spacecraft's motion. A ground antenna transmits a coded signal; the spacecraft returns it. Corrected round-trip time constrains distance, while the Doppler effect of a coherent carrier constrains the line-of-sight relative velocity. NASA's navigation account describes calibration of onboard turnaround delays before launch, calibration of ground delays before use, and corrections for the terrestrial atmosphere. Measurements over multiple stations and epochs constrain the orbit; range and Doppler effect alone do not directly give all transverse motion. Navigators compare a fitted spacecraft trajectory with later tracking and remeasure after propulsion to assess the maneuver. The account identifies radiation pressure and imperfect propulsion as disturbances. This provides practical checks of a dynamical model and its corrections, rather than a laboratory measurement of a universal escape speed. The simple inverse-square potential predicts an unpowered exterior arc; accurate navigation includes additional forces and measurement delays.^[ref]

A spacecraft trajectory is the path traced by a spacecraft's position over time. An unpowered exterior arc can approximate a Kepler orbit; other bodies' gravity, radiation pressure, and propulsion can change it.

Orbit determination infers a body's position and velocity from observations and an equation of motion. It is an inverse problem; a residual of an inverse problem compares an observed readout with its modeled prediction.

For a sphere of radius $r_0$ and uniform mass density ${\rho }_0$, with no surrounding mass and gravitational potential zero at infinity, the spherical shell theorem gives $V_c^2=4\pi G{\rho }_0{r}^2/3$ inside. Matching to the exterior point-mass potential gives $\Phi (r)=-(2\pi G{\rho }_0/3)(3r_0^2-r^2)$. Since $v_e^2=-2\Phi$, the displayed relation follows for $0<r\le r_0$. Exterior shells leave the interior circular speed unchanged but deepen the potential and change the escape velocity; a uniform core alone is insufficient to determine escape.

An equipotential surface is a level set of a scalar potential. Its gradient and the corresponding conservative force are normal to every smooth level surface.

An equipotential curve is a planar level curve of a scalar potential. Where the gradient is nonzero, its tangent is perpendicular to that gradient. In potential flow, it is therefore perpendicular to the velocity field; this distinguishes equipotential curves from streamlines.