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The **classical rocket equation**, or **ideal rocket equation** is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity and can thereby move due to the conservation of momentum. It is credited to Konstantin Tsiolkovsky, who independently derived it and published it in 1903,^{ [1] }^{ [2] } although it had been independently derived and published by William Moore in 1810,^{ [3] } and later published in a separate book in 1813.^{ [4] } Robert Goddard also developed it independently in 1912, and Hermann Oberth derived it independently about 1920.

- History
- Experiment of the Boat by Tsiolkovsky
- Derivation
- Most popular derivation
- Other derivations
- Special relativity
- Terms of the equation
- Delta-v
- Mass fraction
- Effective exhaust velocity
- Applicability
- Examples
- Stages
- See also
- References
- External links

The maximum change of velocity of the vehicle, (with no external forces acting) is:

where:

- is the effective exhaust velocity;
- is the specific impulse in dimension of time;
- is standard gravity;

- is the natural logarithm function;
- is the initial total mass, including propellant, a.k.a. wet mass;
- is the final total mass without propellant, a.k.a. dry mass.

Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g., orbital speed or escape velocity), and a given dry mass , the equation can be solved for the required propellant mass :

The necessary wet mass grows exponentially with the desired delta-v.

The equation is named after Russian scientist Konstantin Tsiolkovsky who independently derived it and published it in his 1903 work.^{ [5] }^{ [2] }

The equation had been derived earlier by the British mathematician William Moore in 1810,^{ [3] } and later published in a separate book in 1813.^{ [4] }

American Robert Goddard independently developed the equation in 1912 when he began his research to improve rocket engines for possible space flight. German engineer Hermann Oberth independently derived the equation about 1920 as he studied the feasibility of space travel.

While the derivation of the rocket equation is a straightforward calculus exercise, Tsiolkovsky is honored as being the first to apply it to the question of whether rockets could achieve speeds necessary for space travel.

In order to understand the principle of rocket propulsion, Konstantin Tsiolkovsky proposed the famous experiment of "the boat". A person is in a boat away from the shore without oars. They want to reach this shore. They notice that the boat is loaded with a certain quantity of stones and have the idea of throwing, one by one and as quickly as possible, these stones in the opposite direction to the bank. Effectively, the quantity of movement of the stones thrown in one direction corresponds to an equal quantity of movement for the boat in the other direction.

Consider the following system:

In the following derivation, "the rocket" is taken to mean "the rocket and all of its unexpended propellant".

Newton's second law of motion relates external forces () to the change in linear momentum of the whole system (including rocket and exhaust) as follows:

where is the momentum of the rocket at time :

and is the momentum of the rocket and exhausted mass at time :

and where, with respect to the observer:

- is the velocity of the rocket at time
- is the velocity of the rocket at time
- is the velocity of the mass added to the exhaust (and lost by the rocket) during time
- is the mass of the rocket at time
- is the mass of the rocket at time

The velocity of the exhaust in the observer frame is related to the velocity of the exhaust in the rocket frame by:

thus,

Solving this yields:

If and are opposite, have the same direction as , are negligible (since ), and using (since ejecting a positive results in a decrease in rocket mass in time),

If there are no external forces then (conservation of linear momentum) and

Assuming that is constant (known as Tsiolkovsky's hypothesis ^{ [2] }), so it is not subject to integration, then the above equation may be integrated as follows:

This then yields

or equivalently

or

or

where is the initial total mass including propellant, the final mass, and the velocity of the rocket exhaust with respect to the rocket (the specific impulse, or, if measured in time, that multiplied by gravity-on-Earth acceleration). If is NOT constant, we might not have rocket equations that are as simple as the above forms. Many rocket dynamics researches were based on the Tsiolkovsky's constant hypothesis.

The value is the total working mass of propellant expended.

(delta v) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (what would be the actual acceleration if external forces were absent). In free space, for the case of acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed. Of course gravity and drag also accelerate the vehicle, and they can add or subtract to the change in velocity experienced by the vehicle. Hence delta-v may not always be the actual change in speed or velocity of the vehicle.

The equation can also be derived from the basic integral of acceleration in the form of force (thrust) over mass. By representing the delta-v equation as the following:

where T is thrust, is the initial (wet) mass and is the initial mass minus the final (dry) mass,

and realising that the integral of a resultant force over time is total impulse, assuming thrust is the only force involved,

The integral is found to be:

Realising that impulse over the change in mass is equivalent to force over propellant mass flow rate (p), which is itself equivalent to exhaust velocity,

the integral can be equated to

Imagine a rocket at rest in space with no forces exerted on it (Newton's First Law of Motion). From the moment its engine is started (clock set to 0) the rocket expels gas mass at a *constant mass flow rate R* (kg/s) and at *exhaust velocity relative to the rocket v _{e}* (m/s). This creates a constant force

Now, the mass of fuel the rocket initially has on board is equal to *m*_{0} – *m _{f}*. For the constant mass flow rate

The rocket equation can also be derived as the limiting case of the speed change for a rocket that expels its fuel in the form of pellets consecutively, as , with an effective exhaust speed such that the mechanical energy gained per unit fuel mass is given by .

In the rocket's center-of-mass frame, if a pellet of mass is ejected at speed and the remaining mass of the rocket is , the amount of energy converted to increase the rocket's and pellet's kinetic energy is

Using momentum conservation in the rocket's frame just prior to ejection, , from which we find

Let be the initial fuel mass fraction on board and the initial fueled-up mass of the rocket. Divide the total mass of fuel into discrete pellets each of mass . The remaining mass of the rocket after ejecting pellets is then . The overall speed change after ejecting pellets is the sum ^{ [6] }

Notice that for large the last term in the denominator and can be neglected to give

where and .

As this Riemann sum becomes the definite integral

since the final remaining mass of the rocket is .

If special relativity is taken into account, the following equation can be derived for a relativistic rocket,^{ [7] } with again standing for the rocket's final velocity (after expelling all its reaction mass and being reduced to a rest mass of ) in the inertial frame of reference where the rocket started at rest (with the rest mass including fuel being initially), and standing for the speed of light in vacuum:

Writing as allows this equation to be rearranged as

Then, using the identity (here "exp" denotes the exponential function; *see also* Natural logarithm as well as the "power" identity at Logarithmic identities) and the identity (*see* Hyperbolic function), this is equivalent to

Delta-*v* (literally "change in velocity"), symbolised as **Δ v** and pronounced

Delta-*v* is produced by reaction engines, such as rocket engines, is proportional to the thrust per unit mass and burn time, and is used to determine the mass of propellant required for the given manoeuvre through the rocket equation.

For multiple manoeuvres, delta-*v* sums linearly.

For interplanetary missions delta-*v* is often plotted on a porkchop plot which displays the required mission delta-*v* as a function of launch date.

In aerospace engineering, the propellant mass fraction is the portion of a vehicle's mass which does not reach the destination, usually used as a measure of the vehicle's performance. In other words, the propellant mass fraction is the ratio between the propellant mass and the initial mass of the vehicle. In a spacecraft, the destination is usually an orbit, while for aircraft it is their landing location. A higher mass fraction represents less weight in a design. Another related measure is the payload fraction, which is the fraction of initial weight that is payload.

The effective exhaust velocity is often specified as a specific impulse and they are related to each other by:

where

- is the specific impulse in seconds,
- is the specific impulse measured in m/s, which is the same as the effective exhaust velocity measured in m/s (or ft/s if g is in ft/s
^{2}), - is the standard gravity, 9.80665 m/s
^{2}(in Imperial units 32.174 ft/s^{2}).

The rocket equation captures the essentials of rocket flight physics in a single short equation. It also holds true for rocket-like reaction vehicles whenever the effective exhaust velocity is constant, and can be summed or integrated when the effective exhaust velocity varies. The rocket equation only accounts for the reaction force from the rocket engine; it does not include other forces that may act on a rocket, such as aerodynamic or gravitational forces. As such, when using it to calculate the propellant requirement for launch from (or powered descent to) a planet with an atmosphere, the effects of these forces must be included in the delta-V requirement (see Examples below). In what has been called "the tyranny of the rocket equation", there is a limit to the amount of payload that the rocket can carry, as higher amounts of propellant increment the overall weight, and thus also increase the fuel consumption.^{ [8] } The equation does not apply to non-rocket systems such as aerobraking, gun launches, space elevators, launch loops, tether propulsion or light sails.

The rocket equation can be applied to orbital maneuvers in order to determine how much propellant is needed to change to a particular new orbit, or to find the new orbit as the result of a particular propellant burn. When applying to orbital maneuvers, one assumes an impulsive maneuver, in which the propellant is discharged and delta-v applied instantaneously. This assumption is relatively accurate for short-duration burns such as for mid-course corrections and orbital insertion maneuvers. As the burn duration increases, the result is less accurate due to the effect of gravity on the vehicle over the duration of the maneuver. For low-thrust, long duration propulsion, such as electric propulsion, more complicated analysis based on the propagation of the spacecraft's state vector and the integration of thrust are used to predict orbital motion.

Assume an exhaust velocity of 4,500 meters per second (15,000 ft/s) and a of 9,700 meters per second (32,000 ft/s) (Earth to LEO, including to overcome gravity and aerodynamic drag).

- Single-stage-to-orbit rocket: = 0.884, therefore 88.4% of the initial total mass has to be propellant. The remaining 11.6% is for the engines, the tank, and the payload.
- Two-stage-to-orbit: suppose that the first stage should provide a of 5,000 meters per second (16,000 ft/s); = 0.671, therefore 67.1% of the initial total mass has to be propellant to the first stage. The remaining mass is 32.9%. After disposing of the first stage, a mass remains equal to this 32.9%, minus the mass of the tank and engines of the first stage. Assume that this is 8% of the initial total mass, then 24.9% remains. The second stage should provide a of 4,700 meters per second (15,000 ft/s); = 0.648, therefore 64.8% of the remaining mass has to be propellant, which is 16.2% of the original total mass, and 8.7% remains for the tank and engines of the second stage, the payload, and in the case of a space shuttle, also the orbiter. Thus together 16.7% of the original launch mass is available for
*all*engines, the tanks, and payload.

In the case of sequentially thrusting rocket stages, the equation applies for each stage, where for each stage the initial mass in the equation is the total mass of the rocket after discarding the previous stage, and the final mass in the equation is the total mass of the rocket just before discarding the stage concerned. For each stage the specific impulse may be different.

For example, if 80% of the mass of a rocket is the fuel of the first stage, and 10% is the dry mass of the first stage, and 10% is the remaining rocket, then

With three similar, subsequently smaller stages with the same for each stage, gives:

and the payload is 10% × 10% × 10% = 0.1% of the initial mass.

A comparable SSTO rocket, also with a 0.1% payload, could have a mass of 11.1% for fuel tanks and engines, and 88.8% for fuel. This would give

If the motor of a new stage is ignited before the previous stage has been discarded and the simultaneously working motors have a different specific impulse (as is often the case with solid rocket boosters and a liquid-fuel stage), the situation is more complicated.

- Delta-v budget
- Jeep problem
- Mass ratio
- Oberth effect - applying delta-v in a gravity well increases the final velocity
- Relativistic rocket
- Reversibility of orbits
- Robert H. Goddard - added terms for gravity and drag in vertical flight
- Spacecraft propulsion
- Stigler’s law of eponymy

**Specific impulse** is a measure of how efficiently a reaction mass engine, such as a rocket using propellant or a jet engine using fuel, generates thrust.

**Bernoulli's principle** is a key concept in fluid dynamics that relates pressure, speed and height. Bernoulli's principle states that an increase in the speed of a parcel of fluid occurs simultaneously with a decrease in either the pressure or the height above a datum. The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book *Hydrodynamica* in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived **Bernoulli's equation** in its usual form.

**Working mass**, also referred to as **reaction mass**, is a mass against which a system operates in order to produce acceleration. In the case of a chemical rocket, for example, the reaction mass is the product of the burned fuel shot backwards to provide propulsion. All acceleration requires an exchange of momentum, which can be thought of as the "unit of movement". Momentum is related to mass and velocity, as given by the formula *P = mv,* where *P* is the momentum, *m* the mass, and *v* the velocity. The velocity of a body is easily changeable, but in most cases the mass is not, which makes it important.

**Delta- v**, symbolized as and pronounced

In classical mechanics, **impulse** is the change in momentum of an object. If the initial momentum of an object is **p**_{1}, and a subsequent momentum is **p**_{2}, the object has received an impulse **J**:

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In aerospace engineering, **mass ratio** is a measure of the efficiency of a rocket. It describes how much more massive the vehicle is with propellant than without; that is, the ratio of the rocket's *wet mass* to its *dry mass*. A more efficient rocket design requires less propellant to achieve a given goal, and would therefore have a lower mass ratio; however, for any given efficiency a higher mass ratio typically permits the vehicle to achieve higher delta-v.

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A **reaction engine** is an engine or motor that produces thrust by expelling reaction mass, in accordance with Newton's third law of motion. This law of motion is commonly paraphrased as: "For every action force there is an equal, but opposite, reaction force."

In astronautics, a **powered flyby**, or **Oberth maneuver**, is a maneuver in which a spacecraft falls into a gravitational well and then uses its engines to further accelerate as it is falling, thereby achieving additional speed. The resulting maneuver is a more efficient way to gain kinetic energy than applying the same impulse outside of a gravitational well. The gain in efficiency is explained by the **Oberth effect**, wherein the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds. In practical terms, this means that the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis, when its orbital velocity is greatest. In some cases, it is even worth spending fuel on slowing the spacecraft into a gravity well to take advantage of the efficiencies of the Oberth effect. The maneuver and effect are named after the person who first described them in 1927, Hermann Oberth, a Transylvanian Saxon physicist and a founder of modern rocketry.

Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation (RTE). However, the RTE is difficult to solve without introducing approximations. A common approximation summarized here is the diffusion approximation. Overall, solutions to the diffusion equation for photon transport are more computationally efficient, but less accurate than Monte Carlo simulations.

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The **Clohessy–Wiltshire equations** describe a simplified model of orbital relative motion, in which the target is in a circular orbit, and the chaser spacecraft is in an elliptical or circular orbit. This model gives a first-order approximation of the chaser's motion in a target-centered coordinate system. It is used to plan the rendezvous of the chaser with the target.

- ↑ К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online here Archived 2011-08-15 at the Wayback Machine in a RARed PDF)
- 1 2 3 Tsiolkovsky, K. "Reactive Flying Machines" (PDF).
- 1 2 Moore, William (1810). "On the Motion of Rockets both in Nonresisting and Resisting Mediums".
*Journal of Natural Philosophy, Chemistry & the Arts*.**27**: 276–285. - 1 2 Moore, William (1813).
*A Treatise on the Motion of Rockets: to which is added, an Essay on Naval Gunnery, in theory and practice, etc*. G. & S. Robinson. - ↑ К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online here Archived 2011-08-15 at the Wayback Machine in a RARed PDF)
- ↑ Blanco, Philip (November 2019). "A discrete, energetic approach to rocket propulsion".
*Physics Education*.**54**(6): 065001. Bibcode:2019PhyEd..54f5001B. doi:10.1088/1361-6552/ab315b. S2CID 202130640. - ↑ Forward, Robert L. "A Transparent Derivation of the Relativistic Rocket Equation" (see the right side of equation 15 on the last page, with
*R*as the ratio of initial to final mass and w as the exhaust velocity, corresponding to*v*_{e}in the notation of this article) - ↑ "The Tyranny of the Rocket Equation".
*NASA.gov*. Archived from the original on 2022-03-06. Retrieved 2016-04-18.

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