ΔV · the physics of DeltaV THRUST

Delta-v and THRUST: the physics of rocket science

A rocket answers two questions. THRUST is how hard it pushes; delta-v is how far that push can take it. This page sets out the conventional physics, the equations every flight dynamics engineer uses, with the sources to read them in full, and then where each idea lives in DeltaV THRUST.

Δ v = v_{e} \ln \frac{m_{0}}{m_{f}}

I · Force

THRUST

A rocket carries its own reaction mass and throws it backwards. By Newton's third law and the conservation of momentum, the vehicle is pushed forward. The THRUST of a rocket nozzle is the momentum flux of the exhaust plus a pressure term at the nozzle exit:

F = ṁ v_{e} + (p_{e} - p_{0}) A_{e}

$ṁ$: mass flow rate of propellant (kg/s)
$v_{e}$: exhaust velocity at the nozzle exit (m/s)
$p_{e}, p_{0}$: exit pressure and ambient pressure (Pa)
$A_{e}$: nozzle exit area (m²)

To leave the pad, THRUST must exceed weight: the THRUST-to-weight ratio $F / (m g_{0})$ must be above 1. Every second spent climbing slowly is lost to gravity drag.

Read: THRUST · NASA Glenn: rocket THRUST summary · NASA Glenn: Beginner's Guide to Rockets

II · Efficiency

Specific impulse

Specific impulse measures how much impulse an engine draws from each unit of propellant. Quoted in seconds, it is the effective exhaust velocity divided by standard gravity:

I_{sp} = \frac{F}{ṁ g_{0}} = \frac{v_{e}}{g_{0}}, g_{0} = 9.80665 m/s²

A kerosene first stage at sea level runs near 300 s; a hydrogen–oxygen upper stage in vacuum runs near 450 s, an exhaust velocity of about 4.41 km/s; ion engines exceed 3,000 s at tiny THRUST. High THRUST and high specific impulse rarely come in the same engine, which is why launch vehicles mix them.

Read: Specific impulse · Exhaust velocity · Standard gravity

III · Reach

The Tsiolkovsky rocket equation

In 1903 Konstantin Tsiolkovsky wrote down the law that governs every rocket. Conserve momentum over an instant in which the vehicle sheds mass $d m$ at exhaust velocity $v_{e}$ :

m d v = - v_{e} d m

Integrate from the full mass $m_{0}$ to the dry mass $m_{f}$ and the rocket equation follows:

Δ v = v_{e} \ln \frac{m_{0}}{m_{f}} = I_{sp} g_{0} \ln \frac{m_{0}}{m_{f}}

The logarithm is the hard truth of spaceflight. Delta-v grows only with the logarithm of the mass ratio, so doubling the propellant buys far less than double the reach. Turned around, the mass ratio a mission demands grows exponentially with its delta-v:

\frac{m_{0}}{m_{f}} = e^{Δ v / v_{e}}

Worked example. Low Earth orbit costs about 9.4 km/s once gravity and drag losses are counted. At 450 s ( $v_{e} \approx 4.413$ km/s) the mass ratio is $e^{9.4 / 4.413} \approx 8.4$ : about 88 % of the vehicle on the pad must be propellant.

Read: Tsiolkovsky rocket equation · Delta-v · Mass ratio · Propellant mass fraction

IV · Structure

Staging: reach is bought with structure

Tanks and engines are dead weight once their propellant is gone. A multistage rocket drops them, and each stage's mass ratio is taken on what remains. The delta-v of the stack is the sum over its stages:

Δ v_{total} = \sum_{i = 1}^{n} v_{e, i} \ln \frac{m_{0, i}}{m_{f, i}}

A single stage whose tanks and engines cap its mass ratio near 8 reaches about 9.2 km/s at 450 s, just short of orbit. Two stages of mass ratio 4 each reach about 12.2 km/s, because the empty first stage never has to be carried to orbit. A bigger tank does not do that; dropping dead weight does.

Read: Multistage rocket · Robert H. Goddard · Hermann Oberth

V · Navigation

Orbits, escape velocity and the delta-v budget

Missions are planned as a delta-v budget: the sum of every burn. Orbital mechanics prices each one from the body's standard gravitational parameter $μ = G M$ . The vis-viva equation gives the speed anywhere on an orbit of semi-major axis $a$ :

v^{2} = μ (\frac{2}{r} - \frac{1}{a})

Two cases follow from it: the circular orbital speed ( $a = r$ ) and the escape velocity ( $a \to \infty$ ):

v_{circ} = \sqrt{\frac{μ}{r}}, v_{esc} = \sqrt{\frac{2 μ}{r}} = \sqrt{2} v_{circ}

At 400 km above Earth a circular orbit runs at 7.67 km/s; escape from Earth's surface needs about 11.2 km/s. Moving between two circular orbits most economically, the Hohmann transfer takes two burns:

Δ v_{1} = \sqrt{\frac{μ}{r_{1}}} (\sqrt{\frac{2 r_{2}}{r_{1} + r_{2}}} - 1), Δ v_{2} = \sqrt{\frac{μ}{r_{2}}} (1 - \sqrt{\frac{2 r_{1}}{r_{1} + r_{2}}})

Worked example. From a 300 km parking orbit to geostationary radius (42,164 km): $Δ v_{1} \approx 2.43$ km/s and $Δ v_{2} \approx 1.47$ km/s, 3.89 km/s in all before any plane change. Burns made deep in a gravity well are worth more (the Oberth effect), and a gravity assist borrows delta-v from a planet.

Read: Delta-v budget · Vis-viva equation · Hohmann transfer orbit · JPL Horizons ephemerides

VI · The limit

The relativistic rocket

Near the speed of light, velocities no longer add; rapidities do. The relativistic rocket equation keeps Tsiolkovsky's logarithm and wraps it in a hyperbolic tangent, so no mass ratio ever reaches $c$ :

Δ v = c \tanh (\frac{v_{e}}{c} \ln \frac{m_{0}}{m_{f}})

For $v_{e} ≪ c$ it reduces exactly to the classical equation.

Read: Relativistic rocket · Physical constants

VII · No return

The event horizon

Set the escape velocity equal to the speed of light and solve for the radius: that is the Schwarzschild radius, the event horizon of a non-rotating black hole, where no delta-v is enough:

r_{s} = \frac{2 G M}{c^{2}}

For the Sun it is about 2.95 km. Matter spiralling in heats into a glowing accretion disk, the burning ring the Event Horizon Telescope imaged around M87* and Sagittarius A*.

Read: Schwarzschild radius · Event horizon · Accretion disk

Instrument

Δv calculator

The rocket equation, live. Enter an engine's specific impulse and the vehicle's masses.

Specific impulse,

I_{sp}

(s) Full mass,

m_{0}

(t) Dry mass,

m_{f}

(t)

Exhaust velocity $v_{e}$: 4,413 m/s
Mass ratio $m_{0} / m_{f}$: 8.333
Delta-v $Δ v$: 9,357 m/s
Propellant fraction: 88.0 %

DeltaV THRUST

THRUST physics: where each law lives in THRUST

The name carries the physics: THRUST is the force. ΔV is the journey. These are analogies of design, not promises of price; every number on this site is read from the chain.

THRUST · $F = ṁ v_{e}$
The token is the propellant, its holders the passengers, the captain the one who plots the burns.
The flight plan →
Staging · $\sum v_{e, i} \ln (m_{0} / m_{f})_{i}$
The bridge is staged the same way: each landing chain carries only what is locked behind it, across seven chains.
The THRUST bridge →
Conservation · $\sum_{c} xTHRUST (c) = outstanding \leq locked$
A lock-and-mint bridge fails one way, by minting what was never locked. The invariant is checked after every step of the rehearsal.
Stardate · releases →
Escape velocity · $\sqrt{2 μ / r}$
The launch tower that does not move: liquidity locked for 100 years, until 2124, that no one can withdraw early.
The 100-year lock →
Instruments · $I_{sp} = v_{e} / g_{0}$
An engineer measures, not guesses. THRUST.oracle prices THRUST from every pool's reserves at one block, to 18 decimals.
THRUST.oracle →
Transfer orbit · $Δ v_{1} + Δ v_{2}$
PULSAR's route is a sequence of transfers through LayerZero, each burn chosen for where its depth is worth it.
The route →
Event horizon · $r_{s} = 2 G M / c^{2}$
Look through the cockpit canopy: the vanishing point is an event horizon with a burning accretion disk, and the thrusters burn aft.
The cockpit →
Network reach · $N \log N$
Reach compounds more slowly than slogans say: Metcalfe's N² is the optimist's count; N log N is the sober one.
The FAQ →

Reference

Constants used on this page

Quantity	Symbol	Value	Source
Speed of light in vacuum	$c$	299,792,458 m/s (exact)	NIST CODATA
Newtonian constant of gravitation	$G$	6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻²	NIST CODATA
Standard gravity	$g_{0}$	9.806 65 m/s² (exact)	Standard gravity
Earth's gravitational parameter	$μ_{\oplus}$	3.986 004 418 × 10¹⁴ m³/s²	Standard gravitational parameter
Earth's equatorial radius	$R_{\oplus}$	6,378.137 km	Orbital speed

Library · 63 links

The top physics links

Wikipedia first, then the agencies, the classrooms and the journals: every one opens in a new tab.

Wikipedia: rocket science

Wikipedia: orbits and gravity

Wikipedia: laws and limits

Wikipedia: the pioneers

Agencies and laboratories

Learn physics

Research and journals

Questions

Delta-v and THRUST: common questions

What is delta-v?

Delta-v (Δv) is the change in velocity a spacecraft can achieve with the propellant it carries. Missions are planned as a delta-v budget, the sum of every burn: about 9.4 km/s to reach low Earth orbit, and about 3.9 km/s more from there to geostationary orbit.

What is THRUST?

THRUST is the force a rocket engine produces by throwing mass backwards: F = ṁvₑ + (pₑ − p₀)Aₑ, the mass flow rate times the exhaust velocity plus the nozzle's pressure term. It is measured in newtons.

What is the Tsiolkovsky rocket equation?

Δv = vₑ ln(m₀/m_f): delta-v equals the exhaust velocity times the natural logarithm of the full mass over the dry mass. Konstantin Tsiolkovsky published it in 1903. Because of the logarithm, the propellant a mission needs grows exponentially with its delta-v.

What is specific impulse?

Specific impulse (Isp) is the impulse an engine produces per unit weight of propellant, in seconds: Isp = vₑ/g₀ with g₀ = 9.80665 m/s². Hydrogen–oxygen engines reach about 450 s in vacuum; ion engines exceed 3,000 s at very low THRUST.

Why do rockets have stages?

Empty tanks and spent engines are dead weight. Dropping them lets each stage start with a fresh mass ratio, so the delta-v of the stages adds up: two stages of mass ratio 4 at 450 s reach about 12.2 km/s, where one stage capped near mass ratio 8 reaches about 9.2 km/s.

What is escape velocity?

The speed at which an object's kinetic energy equals the depth of the gravity well, √(2μ/r), so it never falls back: about 11.2 km/s from Earth's surface. Where it reaches the speed of light, r = 2GM/c², is a black hole's event horizon.

What does ΔV mean in DeltaV THRUST?

The name carries the physics: THRUST is the force, ΔV is the journey. The token is the propellant; the bridge is staged like a rocket, each chain carrying only what is locked behind it; the liquidity is locked for 100 years. These are analogies of design, not promises of price.