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**Thrust** is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that system.^{ [2] } The force applied on a surface in a direction perpendicular or normal to the surface is also called thrust. Force, and thus thrust, is measured using the International System of Units (SI) in newtons (symbol: N), and represents the amount needed to accelerate 1 kilogram of mass at the rate of 1 meter per second per second.^{ [3] } In mechanical engineering, force orthogonal to the main load (such as in parallel helical gears) is referred to as static thrust.

A fixed-wing aircraft propulsion system generates forward thrust when air is pushed in the direction opposite to flight. This can be done by different means such as the spinning blades of a propeller, the propelling jet of a jet engine, or by ejecting hot gases from a rocket engine.^{ [4] } Reverse thrust can be generated to aid braking after landing by reversing the pitch of variable-pitch propeller blades, or using a thrust reverser on a jet engine. Rotary wing aircraft use rotors and thrust vectoring V/STOL aircraft use propellers or engine thrust to support the weight of the aircraft and to provide forward propulsion.

A motorboat propeller generates thrust when it rotates and forces water backwards.

A rocket is propelled forward by a thrust equal in magnitude, but opposite in direction, to the time-rate of momentum change of the exhaust gas accelerated from the combustion chamber through the rocket engine nozzle. This is the exhaust velocity with respect to the rocket, times the time-rate at which the mass is expelled, or in mathematical terms:

Where **T** is the thrust generated (force), is the rate of change of mass with respect to time (mass flow rate of exhaust), and **v** is the velocity of the exhaust gases measured relative to the rocket.

For vertical launch of a rocket the initial thrust at liftoff must be more than the weight.

Each of the three Space Shuttle Main Engines could produce a thrust of 1.8 meganewton, and each of the Space Shuttle's two Solid Rocket Boosters 14.7 MN (3,300,000 lbf ), together 29.4 MN.^{ [5] }

By contrast, the Simplified Aid for EVA Rescue (SAFER) has 24 thrusters of 3.56 N (0.80 lbf) each.^{ [6] }

In the air-breathing category, the AMT-USA AT-180 jet engine developed for radio-controlled aircraft produce 90 N (20 lbf) of thrust.^{ [7] } The GE90-115B engine fitted on the Boeing 777-300ER, recognized by the Guinness Book of World Records as the "World's Most Powerful Commercial Jet Engine," has a thrust of 569 kN (127,900 lbf) until it was surpassed by the GE9X, fitted on the upcoming Boeing 777X, at 609 kN (134,300 lbf).

The power needed to generate thrust and the force of the thrust can be related in a non-linear way. In general, . The proportionality constant varies, and can be solved for a uniform flow, where is the incoming air velocity, is the velocity at the actuator disc, and is the final exit velocity:

Solving for the velocity at the disc, , we then have:

When incoming air is accelerated from a standstill – for example when hovering – then , and we can find:

From here we can see the relationship, finding:

The inverse of the proportionality constant, the "efficiency" of an otherwise-perfect thruster, is proportional to the area of the cross section of the propelled volume of fluid () and the density of the fluid (). This helps to explain why moving through water is easier and why aircraft have much larger propellers than watercraft.

A very common question is how to compare the thrust rating of a jet engine with the power rating of a piston engine. Such comparison is difficult, as these quantities are not equivalent. A piston engine does not move the aircraft by itself (the propeller does that), so piston engines are usually rated by how much power they deliver to the propeller. Except for changes in temperature and air pressure, this quantity depends basically on the throttle setting.

A jet engine has no propeller, so the propulsive power of a jet engine is determined from its thrust as follows. Power is the force (F) it takes to move something over some distance (d) divided by the time (t) it takes to move that distance:^{ [8] }

In case of a rocket or a jet aircraft, the force is exactly the thrust (T) produced by the engine. If the rocket or aircraft is moving at about a constant speed, then distance divided by time is just speed, so power is thrust times speed:^{ [9] }

This formula looks very surprising, but it is correct: the *propulsive power* (or *power available*^{ [10] }) of a jet engine increases with its speed. If the speed is zero, then the propulsive power is zero. If a jet aircraft is at full throttle but attached to a static test stand, then the jet engine produces no propulsive power, however thrust is still produced. The combination piston engine–propeller also has a propulsive power with exactly the same formula, and it will also be zero at zero speed – but that is for the engine–propeller set. The engine alone will continue to produce its rated power at a constant rate, whether the aircraft is moving or not.

Now, imagine the strong chain is broken, and the jet and the piston aircraft start to move. At low speeds:

The piston engine will have constant 100% power, and the propeller's thrust will vary with speed

The jet engine will have constant 100% thrust, and the engine's power will vary with speed

If a powered aircraft is generating thrust T and experiencing drag D, the difference between the two, T − D, is termed the excess thrust. The instantaneous performance of the aircraft is mostly dependent on the excess thrust.

Excess thrust is a vector and is determined as the vector difference between the thrust vector and the drag vector.

The thrust axis for an airplane is the line of action of the total thrust at any instant. It depends on the location, number, and characteristics of the jet engines or propellers. It usually differs from the drag axis. If so, the distance between the thrust axis and the drag axis will cause a moment that must be resisted by a change in the aerodynamic force on the horizontal stabiliser.^{ [11] } Notably, the Boeing 737 MAX, with larger, lower-slung engines than previous 737 models, had a greater distance between the thrust axis and the drag axis, causing the nose to rise up in some flight regimes, necessitating a pitch-control system, MCAS. Early versions of MCAS malfunctioned in flight with catastrophic consequences, leading to the deaths of over 300 people in 2018 and 2019.^{ [12] }^{ [13] }

- Aerodynamic force – Force exerted on a body as it moves through air or gas
- Astern propulsion – Use of a ship's propelling mechanism to develop thrust in a retrograde direction
- Gas turbine engine thrust
- Gimballed thrust – System of thrust vectoring used in rockets (most common in modern rockets)
- Pound (force) – Earth's gravitational pull on a one-pound mass
- "Pound of thrust": thrust (force) required to accelerate one pound at one g

- Stream thrust averaging – Process to convert 3D flow into 1D
- Thrust-to-weight ratio – Dimensionless ratio of thrust to weight of a jet or propeller engine
- Thrust vectoring – Facet of ballistics and aeronautics
- Thrust reversal – Temporary diversion of an aircraft engine's thrust
- Tractive effort – Mechanical engineering term that refers to the amount of traction.

A **centripetal force** is a force that makes a body follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

A **jet engine** is a type of reaction engine, discharging a fast-moving jet of heated gas that generates thrust by jet propulsion. While this broad definition may include rocket, water jet, and hybrid propulsion, the term *jet engine* typically refers to an internal combustion air-breathing jet engine such as a turbojet, turbofan, ramjet, pulse jet, or scramjet. In general, jet engines are internal combustion engines.

In physics and mechanics, **torque** is the rotational analogue of linear force. It is also referred to as the **moment of force**. The symbol for torque is typically , the lowercase Greek letter *tau*. When being referred to as moment of force, it is commonly denoted by M. Just as a linear force is a push or a pull applied to a body, a torque can be thought of as a twist applied to an object with respect to a chosen point; for example, driving a screw uses torque, which is applied by the screwdriver rotating around its axis. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum.

**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.

The **Navier–Stokes equations** are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

In fluid dynamics, **Stokes' law** is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

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**:

**Terminal velocity** is the maximum speed attainable by an object as it falls through a fluid. It is reached when the sum of the drag force (*F _{d}*) and the buoyancy is equal to the downward force of gravity (

In continuum mechanics, the **Froude number** is a dimensionless number defined as the ratio of the flow inertia to the external force field. The Froude number is based on the **speed–length ratio** which he defined as:

**Thrust-to-weight ratio** is a dimensionless ratio of thrust to weight of a rocket, jet engine, propeller engine, or a vehicle propelled by such an engine that is an indicator of the performance of the engine or vehicle.

**Hydraulic head** or **piezometric head** is a specific measurement of liquid pressure above a vertical datum.

**Stokes flow**, also named **creeping flow** or **creeping motion**, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature, this type of flow occurs in the swimming of microorganisms and sperm. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.

In fluid mechanics, **added mass** or **virtual mass** is the inertia added to a system because an accelerating or decelerating body must move some volume of surrounding fluid as it moves through it. Added mass is a common issue because the object and surrounding fluid cannot occupy the same physical space simultaneously. For simplicity this can be modeled as some volume of fluid moving with the object, though in reality "all" the fluid will be accelerated, to various degrees.

The maximal total **range** is the maximum distance an aircraft can fly between takeoff and landing. Powered aircraft range is limited by the aviation fuel energy storage capacity considering both weight and volume limits. Unpowered aircraft range depends on factors such as cross-country speed and environmental conditions. The range can be seen as the cross-country ground speed multiplied by the maximum time in the air. The fuel time limit for powered aircraft is fixed by the available fuel and rate of consumption.

**Ewald summation**, named after Paul Peter Ewald, is a method for computing long-range interactions in periodic systems. It was first developed as the method for calculating the electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system to accurately calculate the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.

In aerospace engineering, concerning aircraft, rocket and spacecraft design, **overall propulsion system efficiency** is the efficiency with which the energy contained in a vehicle's fuel is converted into kinetic energy of the vehicle, to accelerate it, or to replace losses due to aerodynamic drag or gravity. Mathematically, it is represented as , where is the cycle efficiency and is the propulsive efficiency.

The **Cauchy momentum equation** is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.

**Blade element momentum theory** is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor.

**Quantum stochastic calculus** is a generalization of stochastic calculus to noncommuting variables. The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoing measurement, as in quantum trajectories. Just as the Lindblad master equation provides a quantum generalization to the Fokker–Planck equation, quantum stochastic calculus allows for the derivation of quantum stochastic differential equations (QSDE) that are analogous to classical Langevin equations.

**Propeller theory** is the science governing the design of efficient propellers. A propeller is the most common propulsor on ships, and on small aircraft.

- ↑ "Lockheed Martin F-35 Joint Strike Fighter Succeeds in First Vertical Landing".
*Media - Lockheed Martin*. Retrieved 4 April 2024. - ↑ "What is Thrust?".
*www.grc.nasa.gov*. Archived from the original on 14 February 2020. Retrieved 2 April 2020. - ↑ "Force and Motion: Definition, Laws & Formula | StudySmarter".
*StudySmarter UK*. Retrieved 12 October 2022. - ↑ "Newton's Third Law of Motion".
*www.grc.nasa.gov*. Archived from the original on 3 February 2020. Retrieved 2 April 2020. - ↑ "Space Launchers - Space Shuttle".
*www.braeunig.us*. Archived from the original on 6 April 2018. Retrieved 16 February 2018. - ↑ Handley, Patrick M.; Hess, Ronald A.; Robinson, Stephen K. (1 February 2018). "Descriptive Pilot Model for the NASA Simplified Aid for Extravehicular Activity Rescue".
*Journal of Guidance, Control, and Dynamics*.**41**(2): 515–518. Bibcode:2018JGCD...41..515H. doi:10.2514/1.G003131. ISSN 0731-5090. - ↑ "AMT-USA jet engine product information". Archived from the original on 10 November 2006. Retrieved 13 December 2006.
- ↑ Yoon, Joe. "Convert Thrust to Horsepower". Archived from the original on 13 June 2010. Retrieved 1 May 2009.
- ↑ Yechout, Thomas; Morris, Steven.
*Introduction to Aircraft Flight Mechanics*. ISBN 1-56347-577-4. - ↑ Anderson, David; Eberhardt, Scott (2001).
*Understanding Flight*. McGraw-Hill. ISBN 0-07-138666-1. - ↑ Kermode, A.C. (1972)
*Mechanics of Flight*, Chapter 5, 8th edition. Pitman Publishing. ISBN 0273316230 - ↑ "Control system under scrutiny after Ethiopian Airlines crash".
*Al Jazeera*. Archived from the original on 28 April 2019. Retrieved 7 April 2019. - ↑ "What is the Boeing 737 Max Maneuvering Characteristics Augmentation System?".
*The Air Current*. 14 November 2018. Archived from the original on 7 April 2019. Retrieved 7 April 2019.

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