Escape Velocity — Physica

Equivalent forms

v_{\text{esc}} = \sqrt{2gR}

\tfrac{1}{2}mv^2 = \frac{GMm}{R}

A single speed marks the boundary between bound and free — same equation for a marble or a starship.

Symbol	Name	SI unit	Dimension	Range
$v_esc$	Escape velocityoutput Minimum speed to escape gravitational binding from radius R	m/s	LT^-1	0 – 100000
$G$	Gravitational constant Universal gravitational constant	m³·kg⁻¹·s⁻²	M^-1L^3T^-2	6.674e-11 – 6.674e-11
$M$	Body mass Mass of the gravitating body	kg	M	100000000000000000000 – 2e+30
$R$	Radius Distance from the center to the launch point	m	L	100000 – 1000000000

Unit systems

Symbol	SI	CGS	Imperial
$v_esc$	m/s	cm/s	ft/s
$G$	m³·kg⁻¹·s⁻²	cm³·g⁻¹·s⁻²	ft³·slug⁻¹·s⁻²
$M$	kg	g	slug
$R$	m	cm	ft

Where it holds

Valid in the Newtonian regime: v_esc << c and the body is approximately spherical. Ignores atmospheric drag and the rotation of the launch body.

Dimensional analysis

v_esc \to [LT^{-1}]

\sqrt{GM/R} \to \sqrt{[M^{-1}L^{3}T^{-2}][M]/[L]} = \sqrt{[L^{2}T^{-2}]} = [LT^{-1}]

Both sides are

[LT^{-1}] = m/s

.

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Discovery

Isaac Newton · 1687

Derived from Newton's universal gravitation. The concept of a velocity beyond which a projectile escapes was implicit in Principia's cannonball thought experiment.

Try this

How fast must a rocket be moving to leave Earth forever?

Compute the minimum speed needed to escape Earth's gravity from its surface using v_esc = √(2GM/R), with M = 5.972e24 kg and R = 6.371e6 m.

Research status: stable

Real-world applications

Spacecraft trajectory design
Defining black holes via the Schwarzschild radius
Atmospheric retention models for planets
Asteroid and comet sample-return missions

Common misconceptions

Escape velocity does not depend on the projectile's mass
It is the speed needed to escape with no further propulsion — engines can let you 'escape' slower
It does not account for air drag, which raises the practical required initial speed

Experimental verification

Confirmed by every successful interplanetary mission: Voyager, New Horizons, and Mars probes all exceeded Earth's escape velocity

(\sim 11.2\,\mathrm{km/s})

.

Derivation

Set kinetic energy equal to the magnitude of gravitational potential energy:

\tfrac{1}{2}mv^{2} = GMm/R

.

Cancel m, solve for v:

v = \sqrt{2GM/R}

.

Mass of the projectile drops out — escape velocity is universal.

Limiting cases

R \to \infty

⟶

v_esc \to 0

Launching from infinity requires no kinetic energy to remain unbound.

M \to 0

⟶

v_esc \to 0

No gravitating mass means nothing to escape from.

2GM/R \to c^{2}

⟶

v_esc \to c

When required speed approaches light speed, the body forms a black hole (Schwarzschild radius).

What if…

What if I launch from a mountaintop?

R increases slightly, lowering v_esc by a tiny amount. At 5 km altitude, v_esc drops $by \sim 4\,\mathrm{m/s}$ — negligible vs Earth's rotation savings.

What if the body is twice as dense (same mass, half radius)?

v_esc grows $by \sqrt{2} \approx 1.414\times$ . Compact bodies require much higher escape speeds.

What if v_esc exceeds c?

The body is a black hole. The Newtonian formula is no longer valid — general relativity replaces it, but predicts the same Schwarzschild radius $r_s = 2GM/c^{2}$ .

1

Escape from Earth's surface

Given ·

G:: 6.674e-11
M:: 5.972e+24
R:: 6371000

Find · v_esc

Steps

Numerator: $2GM = 2 \times 6.674e-11 \times 5.972e24 = 7.972e14$
Divide by R: $7.972e14 / 6.371e_{6} = 1.251e_{8}$
Take square root: $\sqrt{1.251}e_{8} \approx 11186\,\mathrm{m/s}$

Result ·

v_esc = \sqrt{2 \times 6.674e-11 \times 5.972e24 / 6.371e_{6}} \approx \sqrt{1.251e_{8}} \approx 11186\,\mathrm{m/s} \approx 11.2\,\mathrm{km/s}

2

Escape from the Moon

Given ·

G:: 6.674e-11
M:: 7.342e+22
R:: 1737000

Find · v_esc

Steps

$2GM = 9.797e12$
Divide by R: $9.797e12 / 1.737e_{6} = 5.640e_{6}$
Square root: $\sqrt{5.640}e_{6} \approx 2375\,\mathrm{m/s}$

Result ·

v_esc = \sqrt{2 \times 6.674e-11 \times 7.342e22 / 1.737e_{6}} \approx 2375\,\mathrm{m/s} \approx 2.38\,\mathrm{km/s}

Newton's Law of Universal Gravitation→Kinetic Energy→Kepler's Third Law→