Editing Rocket (section)

==Physics==

===Operation===
{{Main|Rocket engine}}
[[File:A balloon with a tapering nozzle.png|thumb|upright|A balloon with a tapering nozzle. The balloon is pushed by the higher pressure at the top than found around the inside of the nozzle.]]

The [[Reaction (physics)|effect]] of the combustion of propellant in the rocket engine is to increase the internal energy of the resulting gases, utilizing the stored chemical energy in the fuel.{{citation needed|date=February 2016}} As the internal energy increases, pressure increases, and a nozzle is used to convert this energy into a directed kinetic energy. This produces thrust against the ambient environment to which these gases are released.{{citation needed|date=February 2016}} The ideal direction of motion of the exhaust is in the direction so as to cause thrust. At the top end of the combustion chamber the hot, energetic gas fluid cannot move forward, and so, it pushes upward against the top of the rocket engine's [[combustion chamber]]. As the combustion gases approach the exit of the combustion chamber, they increase in speed. The effect of the [[Rocket engine nozzle|convergent]] part of the rocket engine nozzle on the high pressure fluid of combustion gases, is to cause the gases to accelerate to high speed. The higher the speed of the gases, the lower the pressure of the gas ([[Bernoulli's principle]] or [[conservation of energy]]) acting on that part of the combustion chamber. In a properly designed engine, the flow will reach Mach 1 at the throat of the nozzle. At which point the speed of the flow increases. Beyond the throat of the nozzle, a bell shaped expansion part of the engine allows the gases that are expanding to push against that part of the rocket engine. Thus, the bell part of the nozzle gives additional thrust. Simply expressed, for every action there is an equal and opposite reaction, according to [[Newton's third law]] with the result that the exiting gases produce the reaction of a force on the rocket causing it to accelerate the rocket.<ref>{{cite book|last1=Warren|first1=J. W.|title=Understanding force: an account of some aspects of teaching the idea of force in school, college and university courses in engineering, mathematics and science|date=1979|publisher=Murray|location=London|isbn=978-0-7195-3564-2|pages=[https://archive.org/details/understandingfor0000warr/page/37 37–38]|url=https://archive.org/details/understandingfor0000warr/page/37}}</ref>{{#tag:ref|"If you have ever seen a big fire hose spraying water, you may have noticed that it takes a lot of strength to hold the hose (sometimes you will see two or three firefighters holding the hose). The hose is acting like a rocket engine. The hose is throwing water in one direction, and the firefighters are using their strength and weight to counteract the reaction. If they were to let go of the hose, it would thrash around with tremendous force. If the firefighters were all standing on skateboards, the hose would propel them backward at great speed!"<ref>The confusion is illustrated in {{cite web |url=http://science.howstuffworks.com/rocket.htm |title=How Rocket Engines Work |last=Brain |first=Marshall |work=Howstuffworks.com |date=April 2000 |access-date=22 August 2022}}</ref>|group=nb}}

[[File:Rocket thrust.svg|thumb|right|Rocket thrust is caused by pressures acting on both the combustion chamber and nozzle]]
In a closed chamber, the pressures are equal in each direction and no acceleration occurs. If an opening is provided in the bottom of the chamber then the pressure is no longer acting on the missing section. This opening permits the exhaust to escape. The remaining pressures give a resultant thrust on the side opposite the opening, and these pressures are what push the rocket along.

The shape of the nozzle is important. Consider a balloon propelled by air coming out of a tapering nozzle. In such a case the combination of air pressure and viscous friction is such that the nozzle does not push the balloon but is ''pulled'' by it.<ref>{{cite book|last1=Warren|first1=J. W.|title=Understanding force: an account of some aspects of teaching the idea of force in school, college and university courses in engineering, mathematics and science|date=1979|publisher=Murray|location=London|isbn=978-0-7195-3564-2|page=[https://archive.org/details/understandingfor0000warr/page/28 28]|url=https://archive.org/details/understandingfor0000warr/page/28}}</ref> Using a convergent/divergent nozzle gives more force since the exhaust also presses on it as it expands outwards, roughly doubling the total force. If propellant gas is continuously added to the chamber then these pressures can be maintained for as long as propellant remains. Note that in the case of liquid propellant engines, the pumps moving the propellant into the combustion chamber must maintain a pressure larger than the combustion chamber—typically on the order of 100 atmospheres.<ref name="RPE7"/>

As a side effect, these pressures on the rocket also act on the exhaust in the opposite direction and accelerate this exhaust to very high speeds (according to [[Newton's third law]]).<ref name="RPE7"/> From the principle of [[conservation of momentum]] the speed of the exhaust of a rocket determines how much momentum increase is created for a given amount of propellant. This is called the rocket's ''[[specific impulse]]''.<ref name="RPE7"/> Because a rocket, propellant and exhaust in flight, without any external perturbations, may be considered as a closed system, the total momentum is always constant. Therefore, the faster the net speed of the exhaust in one direction, the greater the speed of the rocket can achieve in the opposite direction. This is especially true since the rocket body's mass is typically far lower than the final total exhaust mass.

===Forces on a rocket in flight===
[[File:Rktfor.gif|thumb|upright|Forces on a rocket in flight]]
The general study of the [[force]]s on a rocket is part of the field of [[ballistics]]. Spacecraft are further studied in the subfield of [[astrodynamics]].

Flying rockets are primarily affected by the following:<ref>{{cite web |url=http://www.grc.nasa.gov/WWW/K-12/VirtualAero/BottleRocket/airplane/rktfor.html |title=Four forces on a model rocket |publisher=NASA |date=2000-09-19 |access-date=2012-12-10 |url-status=dead |archive-url=https://web.archive.org/web/20121129204215/http://www.grc.nasa.gov/WWW/k-12/VirtualAero/BottleRocket/airplane/rktfor.html |archive-date=2012-11-29 }}</ref>
* [[Thrust]] from the engine(s)
* [[Gravity]] from [[celestial bodies]]
* [[Drag (physics)|Drag]] if moving in atmosphere
* [[Lift (force)|Lift]]; usually relatively small effect except for [[rocket-powered aircraft]]

In addition, the [[centrifugal force (fictitious)|inertia and centrifugal pseudo-force]] can be significant due to the path of the rocket around the center of a celestial body; when high enough speeds in the right direction and altitude are achieved a stable [[orbit]] or [[escape velocity]] is obtained.

These forces, with a stabilizing tail (the ''[[empennage]]'') present will, unless deliberate control efforts are made, naturally cause the vehicle to follow a roughly [[parabola|parabolic]] trajectory termed a [[gravity turn]], and this trajectory is often used at least during the initial part of a launch. (This is true even if the rocket engine is mounted at the nose.) Vehicles can thus maintain low or even zero [[angle of attack]], which minimizes transverse [[stress (physics)|stress]] on the [[launch vehicle]], permitting a weaker, and hence lighter, launch vehicle.<ref name=space-sourcebook>{{cite book|first1=Samuel|last1=Glasstone|title=Sourcebook on the Space Sciences|url=https://books.google.com/books?id=K6k0AAAAMAAJ|publisher=D. Van Nostrand Co.|date= 1965|access-date=28 May 2016|page=209|oclc=232378|url-status=live|archive-url=https://web.archive.org/web/20171119163047/https://books.google.com/books?id=K6k0AAAAMAAJ|archive-date=19 November 2017}}</ref><ref name=thesis>{{Cite thesis|first=David W. |last=Callaway |title=Coplanar Air Launch with Gravity-Turn Launch Trajectories |type=Master's thesis |publisher=Air Force Institute of Technology |date=March 2004 |url=https://scholar.afit.edu/etd/3922/ |page = 2 }}</ref>
{{clear}}

====Drag====
{{main|Drag (physics)|Gravity drag|Aerodynamic drag}}
Drag is a force opposite to the direction of the rocket's motion relative to any air it is moving through. This slows the speed of the vehicle and produces structural loads. The deceleration forces for fast-moving rockets are calculated using the [[drag equation]].

Drag can be minimised by an aerodynamic [[nose cone]] and by using a shape with a high [[ballistic coefficient]] (the "classic" rocket shape—long and thin), and by keeping the rocket's [[angle of attack]] as low as possible.

During a launch, as the vehicle speed increases, and the atmosphere thins, there is a point of maximum aerodynamic drag called [[max Q]]. This determines the minimum aerodynamic strength of the vehicle, as the rocket must avoid [[buckling]] under these forces.<ref name=maxq>{{cite web |url=http://www.aerospaceweb.org/question/aerodynamics/q0025.shtml |title=Space Shuttle Max-Q |publisher=Aerospaceweb |date=2001-05-06 |access-date=2012-12-10}}</ref>

====Net thrust====
{{For|a more detailed model of the net thrust of a rocket engine that includes the effect of atmospheric pressure|Rocket engine#Net thrust}}
[[File:Rocket nozzle expansion.svg|thumb|upright|[[Rocket engine#Nozzle|A rocket jet shape]] varies based on external air pressure. From top to bottom:{{unbulleted list|Underexpanded|Ideally expanded|Overexpanded|Grossly overexpanded}}]]

A typical rocket engine can handle a significant fraction of its own mass in propellant each second, with the propellant leaving the nozzle at several kilometres per second. This means that the [[thrust-to-weight ratio]] of a rocket engine, and often the entire vehicle can be very high, in extreme cases over 100. This compares with other jet propulsion engines that can exceed 5 for some of the better<ref>{{cite web |url=http://www.geae.com/engines/military/j85/index.html |title=General Electric J85 |publisher=Geae.com |date=2012-09-07 |access-date=2012-12-10 |url-status=dead |archive-url=https://web.archive.org/web/20110722155949/http://www.geae.com/engines/military/j85/index.html |archive-date=2011-07-22 }}</ref> engines.<ref>{{cite web |url=http://www.thrustssc.com/thrustssc/Club/Secure/Arfons_Last_Stand.html |title=Mach 1 Club |publisher=Thrust SSC |access-date=2016-05-28 |url-status=dead |archive-url=https://web.archive.org/web/20160617103717/http://www.thrustssc.com/thrustssc/Club/Secure/Arfons_Last_Stand.html |archive-date=2016-06-17 }}</ref>

The net thrust of a rocket is

{{block indent|<math>F_n = \dot{m}\;v_{e},</math><ref name="RPE7"/>{{rp|2–14}}}}

where

{{block indent|<math> \dot{m} =\,</math>propellant flow (kg/s or lb/s)}}
{{block indent|<math>v_{e} =\,</math>the [[effective exhaust velocity]] (m/s or ft/s).}}

The effective exhaust velocity <math>v_{e}</math> is more or less the speed the exhaust leaves the vehicle, and in the vacuum of space, the effective exhaust velocity is often equal to the actual average exhaust speed along the thrust axis. However, the effective exhaust velocity allows for various losses, and notably, is reduced when operated within an atmosphere.

The rate of propellant flow through a rocket engine is often deliberately varied over a flight, to provide a way to control the thrust and thus the airspeed of the vehicle. This, for example, allows minimization of aerodynamic losses<ref name=maxq/> and can limit the increase of [[g-force|''g''-forces]] due to the reduction in propellant load.

===Total impulse===
{{Main|Impulse (physics)}}
Impulse is defined as a force acting on an object over time, which in the absence of opposing forces (gravity and aerodynamic drag), changes the [[momentum]] (integral of mass and velocity) of the object. As such, it is the best performance class (payload mass and terminal velocity capability) indicator of a rocket, rather than takeoff thrust, mass, or "power". The total impulse of a rocket (stage) burning its propellant is:<ref name="RPE7"/>{{rp|27}}

{{block indent|<math>I = \int F dt</math>}}

When there is fixed thrust, this is simply:

{{block indent|<math>I = F t\;</math>}}

The total impulse of a multi-stage rocket is the sum of the impulses of the individual stages.
{{Clear}}

===Specific impulse===
{{Main|specific impulse}}
{{Specific impulse|align=right}}
As can be seen from the thrust equation, the effective speed of the exhaust controls the amount of thrust produced from a particular quantity of fuel burnt per second.

An equivalent measure, the net impulse per weight unit of propellant expelled, is called [[Specific impulse|specific Impulse]], <math>I_{sp}</math>, and this is one of the most important figures that describes a rocket's performance. It is defined such that it is related to the effective exhaust velocity by:

{{block indent|<math>v_e = I_{sp} \cdot g_0</math><ref name="RPE7"/>{{rp|29}}}}

where:
{{block indent|<math>I_{sp}</math> has units of seconds}}
{{block indent|<math>g_0</math> is the acceleration at the surface of the Earth}}

Thus, the greater the specific impulse, the greater the net thrust and performance of the engine. <math>I_{sp}</math> is determined by measurement while testing the engine. In practice the effective exhaust velocities of rockets varies but can be extremely high, ~4500&nbsp;m/s, about 15 times the sea level speed of sound in air.

===Delta-v (rocket equation)===
{{Main|Tsiolkovsky rocket equation}}
[[File:Delta-Vs for inner Solar System.svg|thumb|upright|A map of approximate [[Delta-v]]'s around the Solar System between Earth and [[Mars]]<ref>{{cite web |url=http://www.pma.caltech.edu/~chirata/deltav.html |title=table of cislunar/mars delta-vs |archive-url=https://web.archive.org/web/20070701211813/http://www.pma.caltech.edu/~chirata/deltav.html |archive-date=2007-07-01}}</ref><ref>{{cite web |url=http://www.strout.net/info/science/delta-v/intro.html |title=cislunar delta-vs |publisher=Strout.net |access-date=2012-12-10 |url-status=live |archive-url=https://web.archive.org/web/20000312041150/http://www.strout.net/info/science/delta-v/intro.html |archive-date=2000-03-12 }}</ref>]]

The [[delta-v]] capacity of a rocket is the theoretical total change in velocity that a rocket can achieve without any external interference (without air drag or gravity or other forces).

When <math>v_e</math> is constant, the delta-v that a rocket vehicle can provide can be calculated from the [[Tsiolkovsky rocket equation]]:<ref>{{cite web |url=http://www.projectrho.com/rocket/rocket3c.html |title=Choose Your Engine |publisher=Projectrho.com |date=2012-06-01 |access-date=2012-12-10 |url-status=live |archive-url=https://web.archive.org/web/20100529042131/http://www.projectrho.com/rocket/rocket3c.html |archive-date=2010-05-29 }}</ref>

:<math>\Delta v\ = v_e \ln \frac {m_0} {m_1}</math>

where:
{{block indent|<math>m_0</math> is the initial total mass, including propellant, in kg (or lb)}}
{{block indent|<math>m_1</math> is the final total mass in kg (or lb)}}
{{block indent|<math>v_e</math> is the effective exhaust velocity in m/s (or ft/s)}}
{{block indent|<math>\Delta v\ </math> is the delta-v in m/s (or ft/s)}}

When launched from the Earth practical delta-vs for a single rockets carrying payloads can be a few km/s. Some theoretical designs have rockets with delta-vs over 9&nbsp;km/s.

The required delta-v can also be calculated for a particular manoeuvre; for example the delta-v to launch from the surface of the Earth to [[low Earth orbit]] is about 9.7&nbsp;km/s, which leaves the vehicle with a sideways speed of about 7.8&nbsp;km/s at an altitude of around 200&nbsp;km. In this manoeuvre about 1.9&nbsp;km/s is lost in [[air drag]], [[gravity drag]] and [[potential energy|gaining altitude]].

The ratio <math>\frac {m_0} {m_1}</math> is sometimes called the ''mass ratio''.

===Mass ratios===
{{Main|Mass ratio}}
[[File:Tsiolkovsky rocket equation.svg|thumb|The Tsiolkovsky rocket equation gives a relationship between the mass ratio and the final velocity in multiples of the exhaust speed]]

Almost all of a launch vehicle's mass consists of propellant.<ref>{{cite web |url=http://www-istp.gsfc.nasa.gov/stargaze/Srockhis.htm |title=The Evolution of Rockets |publisher=Istp.gsfc.nasa.gov |access-date=2012-12-10 |url-status=dead |archive-url=https://web.archive.org/web/20130108191716/http://www-istp.gsfc.nasa.gov/stargaze/Srockhis.htm |archive-date=2013-01-08 }}</ref> Mass ratio is, for any 'burn', the ratio between the rocket's initial mass and its final mass.<ref>{{cite web |url=http://exploration.grc.nasa.gov/education/rocket/rktwtp.html |title=Rocket Mass Ratios |publisher=Exploration.grc.nasa.gov |access-date=2012-12-10 |url-status=dead |archive-url=https://web.archive.org/web/20130216063603/http://exploration.grc.nasa.gov/education/rocket/rktwtp.html |archive-date=2013-02-16 }}</ref> Everything else being equal, a high mass ratio is desirable for good performance, since it indicates that the rocket is lightweight and hence performs better, for essentially the same reasons that low weight is desirable in sports cars.

Rockets as a group have the highest [[thrust-to-weight ratio]] of any type of engine; and this helps vehicles achieve high [[mass ratio]]s, which improves the performance of flights. The higher the ratio, the less engine mass is needed to be carried. This permits the carrying of even more propellant, enormously improving the delta-v. Alternatively, some rockets such as for rescue scenarios or racing carry relatively little propellant and payload and thus need only a lightweight structure and instead achieve high accelerations. For example, the Soyuz escape system can produce 20&nbsp;''g''.<ref name=soyuzt/>

Achievable mass ratios are highly dependent on many factors such as propellant type, the design of engine the vehicle uses, structural safety margins and construction techniques.

The highest mass ratios are generally achieved with liquid rockets, and these types are usually used for [[orbital launch vehicle]]s, a situation which calls for a high delta-v. Liquid propellants generally have densities similar to water (with the notable exceptions of [[liquid hydrogen]] and [[Liquid methane rocket fuel|liquid methane]]), and these types are able to use lightweight, low pressure tanks and typically run high-performance [[turbopumps]] to force the propellant into the combustion chamber.

Some notable mass fractions are found in the following table (some aircraft are included for comparison purposes):

{| class="wikitable"
|-
! Vehicle
! Takeoff mass
! Final mass
! [[Mass ratio]]
! [[Propellant mass fraction|Mass fraction]]
|-
| [[Ariane 5]] (vehicle + payload)
| 746,000&nbsp;kg <ref name="ariane">{{cite web| url = http://astronautix.com/lvs/ariane5g.htm| archive-url = https://web.archive.org/web/20031225090030/http://www.astronautix.com/lvs/ariane5g.htm| url-status = dead| archive-date = December 25, 2003| title = Astronautix – Ariane 5g}}</ref> (~1,645,000&nbsp;lb)
| 2,700&nbsp;kg + 16,000&nbsp;kg<ref name="ariane"/> (~6,000&nbsp;lb + ~35,300&nbsp;lb)
| 39.9
| 0.975
|-
| [[Titan 23G]] first stage
| 117,020&nbsp;kg (258,000&nbsp;lb)
| 4,760&nbsp;kg (10,500&nbsp;lb)
| 24.6
| 0.959
|-
| [[Saturn V]]
| 3,038,500&nbsp;kg<ref name="saturnv">{{cite web| url = http://astronautix.com/lvs/saturnv.htm| archive-url = https://web.archive.org/web/20020228155213/http://www.astronautix.com/lvs/saturnv.htm| url-status = dead| archive-date = February 28, 2002| title = Astronautix – Saturn V}}</ref> (~6,700,000&nbsp;lb)
| 13,300&nbsp;kg + 118,000&nbsp;kg<ref name="saturnv"/> (~29,320&nbsp;lb + ~260,150&nbsp;lb)
| 23.1
| 0.957
|-
| [[Space Shuttle]] (vehicle + payload)
| 2,040,000&nbsp;kg (~4,500,000&nbsp;lb)
| 104,000&nbsp;kg + 28,800&nbsp;kg (~230,000&nbsp;lb + ~63,500&nbsp;lb)
| 15.4
| 0.935
|-
| [[Saturn 1B]] (stage only)
| 448,648&nbsp;kg<ref name="saturn">{{cite web| url = http://astronautix.com/lvs/saturnib.htm| archive-url = https://web.archive.org/web/20020305051145/http://www.astronautix.com/lvs/saturnib.htm| url-status = dead| archive-date = March 5, 2002| title = Astronautix – Saturn IB}}</ref> (989,100&nbsp;lb)
| 41,594&nbsp;kg<ref name="saturn"/> (91,700&nbsp;lb)
| 10.7
| 0.907
|-
| [[Virgin Atlantic GlobalFlyer]]
| 10,024.39&nbsp;kg (22,100&nbsp;lb)
| 1,678.3&nbsp;kg (3,700&nbsp;lb)
| 6.0
| 0.83
|-
| [[V-2 rocket|V-2]]
| 13,000&nbsp;kg (~28,660&nbsp;lb) (12.8 ton)
| 
| 3.85
| 0.74 <ref>{{cite web| url = http://www.astronautix.com/lvs/v2.htm| archive-url = https://web.archive.org/web/20020302013547/http://www.astronautix.com/lvs/v2.htm| url-status = dead| archive-date = March 2, 2002| title = Astronautix-V-2}}</ref>
|-
| [[X-15]]
| 15,420&nbsp;kg (34,000&nbsp;lb)
| 6,620&nbsp;kg (14,600&nbsp;lb)
| 2.3
| 0.57<ref name="rlv"/>
|-
| [[Concorde]]
| ~181,000&nbsp;kg (400,000&nbsp;lb <ref name="rlv">{{cite web |url=http://mae.ucdavis.edu/faculty/sarigul/aiaa2001-4619.pdf |title=AIAA2001-4619 RLVs |access-date=2019-02-19 |archive-url=https://web.archive.org/web/20131206012152/http://mae.ucdavis.edu/faculty/sarigul/aiaa2001-4619.pdf |archive-date=2013-12-06 |url-status=dead }}</ref>)
| 
| 2
| 0.5<ref name="rlv"/>
|-
| [[Boeing 747]]
| ~363,000&nbsp;kg (800,000&nbsp;lb<ref name="rlv"/>)
| 
| 2
| 0.5<ref name="rlv"/>
|}

===Staging===
{{Main|Multistage rocket}}
[[File:Artistsconcept separation.jpg|thumb|Spacecraft staging involves dropping off unnecessary parts of the rocket to reduce mass]]
[[File:Ap6-68-HC-191.jpg|thumb|[[Apollo 6]] while dropping the interstage ring]]
Thus far, the required velocity (delta-v) to achieve orbit has been unattained by any single rocket because the [[propellant]], tankage, structure, [[guidance system|guidance]], valves and engines and so on, take a particular minimum percentage of take-off mass that is too great for the propellant it carries to achieve that delta-v carrying reasonable payloads. Since [[Single-stage-to-orbit]] has so far not been achievable, orbital rockets always have more than one stage.

For example, the first stage of the Saturn V, carrying the weight of the upper stages, was able to achieve a [[mass ratio]] of about 10, and achieved a specific impulse of 263 seconds. This gives a delta-v of around 5.9&nbsp;km/s whereas around 9.4&nbsp;km/s delta-v is needed to achieve orbit with all losses allowed for.

This problem is frequently solved by [[staging (rocketry)|staging]]—the rocket sheds excess weight (usually empty tankage and associated engines) during launch. Staging is either ''serial'' where the rockets light after the previous stage has fallen away, or ''parallel'', where rockets are burning together and then detach when they burn out.<ref name="NASAstaging">{{cite web |author=NASA |url=https://spaceflightsystems.grc.nasa.gov/education/rocket/rktstage.html |title=Rocket staging |access-date=2016-05-28 |publisher=NASA |work=Beginner's Guide to Rockets |year=2006 |url-status=dead |archive-url=https://web.archive.org/web/20160602123849/https://spaceflightsystems.grc.nasa.gov/education/rocket/rktstage.html |archive-date=2016-06-02 }}</ref>

The maximum speeds that can be achieved with staging is theoretically limited only by the speed of light. However the payload that can be carried goes down geometrically with each extra stage needed, while the additional delta-v for each stage is simply additive.

===Acceleration and thrust-to-weight ratio===
{{Main|thrust-to-weight ratio}}
From Newton's second law, the acceleration, <math>a</math>, of a vehicle is simply:

{{block indent|<math>a = \frac {F_n} {m}</math>}}
where {{mvar|m}} is the instantaneous mass of the vehicle and <math>F_n</math> is the net force acting on the rocket (mostly thrust, but air drag and other forces can play a part).

As the remaining propellant decreases, rocket vehicles become lighter and their acceleration tends to increase until the propellant is exhausted. This means that much of the speed change occurs towards the end of the burn when the vehicle is much lighter.<ref name="RPE7"/> However, the thrust can be throttled to offset or vary this if needed. Discontinuities in acceleration also occur when stages burn out, often starting at a lower acceleration with each new stage firing.

Peak accelerations can be increased by designing the vehicle with a reduced mass, usually achieved by a reduction in the fuel load and tankage and associated structures, but obviously this reduces range, delta-v and burn time. Still, for some applications that rockets are used for, a high peak acceleration applied for just a short time is highly desirable.

The minimal mass of vehicle consists of a rocket engine with minimal fuel and structure to carry it. In that case the [[thrust-to-weight ratio]]{{#tag:ref|"thrust-to-weight ratio {{math|{{var|F}}/{{var|W}}{{sub|{{var|g}}}}}} is a dimensionless parameter that is identical to the acceleration of the rocket propulsion system (expressed in multiples of {{math|{{var|g}}{{sub|0}}}}) ... in a gravity-free vacuum"<ref name="RPE7"/>{{rp|442}}|group=nb}} of the rocket engine limits the maximum acceleration that can be designed. It turns out that rocket engines generally have truly excellent thrust to weight ratios (137 for the [[NK-33]] engine;<ref>{{cite web |url=http://www.astronautix.com/engines/nk33.htm |title=Astronautix NK-33 entry |publisher=Astronautix.com |date=2006-11-08 |access-date=2012-12-10 |url-status=dead |archive-url=https://web.archive.org/web/20020625124013/http://www.astronautix.com/engines/nk33.htm |archive-date=2002-06-25 }}</ref> some solid rockets are over 1000<ref name="RPE7"/>{{rp|442}}), and nearly all really [[g-force|high-g]] vehicles employ or have employed rockets.

The high accelerations that rockets naturally possess means that rocket vehicles are often capable of [[vertical takeoff]], and in some cases, with suitable guidance and control of the engines, also [[VTVL|vertical landing]]. For these operations to be done it is necessary for a vehicle's engines to provide more than the local [[gravitational acceleration]].

===Energy===

====Energy efficiency====
{{Main|propulsive efficiency}}
[[File:Atlantis taking off on STS-27.jpg|thumb|{{OV|104}} during launch phase]]

The energy density of a typical rocket propellant is often around one-third that of conventional hydrocarbon fuels; the bulk of the mass is (often relatively inexpensive) oxidizer. Nevertheless, at take-off the rocket has a great deal of energy in the fuel and oxidizer stored within the vehicle. It is of course desirable that as much of the energy of the propellant end up as [[kinetic energy|kinetic]] or [[potential energy]] of the body of the rocket as possible.

Energy from the fuel is lost in air drag and [[gravity drag]] and is used for the rocket to gain altitude and speed. However, much of the lost energy ends up in the exhaust.<ref name="RPE7"/>{{rp|37–38}}

In a chemical propulsion device, the engine efficiency is simply the ratio of the kinetic power of the exhaust gases and the power available from the chemical reaction:<ref name="RPE7"/>{{rp|37–38}}

{{block indent|<math>\eta_c= \frac {\frac {1} {2}\dot{m}v_e^2} {\eta_{combustion} P_{chem} }</math>}}

100% efficiency within the engine ([[Heat engine#Efficiency|engine efficiency]] <math>\eta_c = 100\%</math>) would mean that all the heat energy of the combustion products is converted into kinetic energy of the jet. This is not possible, but the near-adiabatic [[rocket engine nozzle|high expansion ratio nozzles]] that can be used with rockets come surprisingly close: when the nozzle expands the gas, the gas is cooled and accelerated, and an energy efficiency of up to 70% can be achieved. Most of the rest is heat energy in the exhaust that is not recovered.<ref name="RPE7"/>{{rp|37–38}} The high efficiency is a consequence of the fact that rocket combustion can be performed at very high temperatures and the gas is finally released at much lower temperatures, and so giving good [[Carnot efficiency]].

However, engine efficiency is not the whole story. In common with the other [[jet engine|jet-based engines]], but particularly in rockets due to their high and typically fixed exhaust speeds, rocket vehicles are extremely inefficient at low speeds irrespective of the engine efficiency. The problem is that at low speeds, the exhaust carries away a huge amount of [[kinetic energy]] rearward. This phenomenon is termed [[propulsive efficiency]] (<math>\eta_p</math>).<ref name="RPE7"/>{{rp|37–38}}

However, as speeds rise, the resultant exhaust speed goes down, and the overall vehicle energetic efficiency rises, reaching a peak of around 100% of the engine efficiency when the vehicle is travelling exactly at the same speed that the exhaust is emitted. In this case the exhaust would ideally stop dead in space behind the moving vehicle, taking away zero energy, and from conservation of energy, all the energy would end up in the vehicle. The efficiency then drops off again at even higher speeds as the exhaust ends up traveling forwards – trailing behind the vehicle.

[[File:Average propulsive efficiency of rockets.png|thumb|Plot of instantaneous propulsive efficiency (blue) and overall efficiency for a rocket accelerating from rest (red) as percentages of the engine efficiency]]From these principles it can be shown that the propulsive efficiency <math>\eta_p</math> for a rocket moving at speed <math>u</math> with an exhaust velocity <math>c</math> is:
{{block indent|<math>\eta_p= \frac {2 \frac {u} {c}} {1 + ( \frac {u} {c} )^2 }</math><ref name="RPE7"/>{{rp|37–38}}}}

And the overall (instantaneous) energy efficiency <math>\eta</math> is:
{{block indent|<math>\eta= \eta_p \eta_c</math>}}

For example, from the equation, with an <math>\eta_c</math> of 0.7, a rocket flying at Mach 0.85 (which most aircraft cruise at) with an exhaust velocity of Mach 10, would have a predicted overall energy efficiency of 5.9%, whereas a conventional, modern, air-breathing jet engine achieves closer to 35% efficiency. Thus a rocket would need about 6x more energy; and allowing for the specific energy of rocket propellant being around one third that of conventional air fuel, roughly 18x more mass of propellant would need to be carried for the same journey. This is why rockets are rarely if ever used for general aviation.

Since the energy ultimately comes from fuel, these considerations mean that rockets are mainly useful when a very high speed is required, such as [[ICBM]]s or [[orbital spaceflight|orbital launch]]. For example, [[NASA]]'s [[Space Shuttle]] fired its engines for around 8.5 minutes, consuming 1,000&nbsp;tonnes of solid propellant (containing 16% aluminium) and an additional 2,000,000&nbsp;litres of liquid propellant (106,261&nbsp;kg of [[liquid hydrogen]] fuel) to lift the 100,000&nbsp;kg vehicle (including the 25,000&nbsp;kg payload) to an altitude of 111&nbsp;km and an orbital [[velocity]] of 30,000&nbsp;km/h. At this altitude and velocity, the vehicle had a kinetic energy of about 3&nbsp;TJ and a potential energy of roughly 200&nbsp;GJ. Given the initial energy of 20&nbsp;TJ,{{#tag:ref|The energy density is 31MJ per kg for aluminum and 143&nbsp;MJ/kg for liquid hydrogen, this means that the vehicle consumed around 5&nbsp;TJ of solid propellant and 15&nbsp;TJ of hydrogen fuel.|group=nb}} the Space Shuttle was about 16% energy efficient at launching the orbiter.

Thus jet engines, with a better match between speed and jet exhaust speed (such as [[turbofans]]—in spite of their worse <math>\eta_c</math>)—dominate for subsonic and supersonic atmospheric use, while rockets work best at hypersonic speeds. On the other hand, rockets serve in many short-range ''relatively'' low speed military applications where their low-speed inefficiency is outweighed by their extremely high thrust and hence high accelerations.

====Oberth effect====
{{Main|Oberth effect}}
One subtle feature of rockets relates to energy. A rocket stage, while carrying a given load, is capable of giving a particular [[delta-v]]. This delta-v means that the speed increases (or decreases) by a particular amount, independent of the initial speed. However, because [[kinetic energy]] is a square law on speed, this means that the faster the rocket is travelling before the burn the more [[orbital energy]] it gains or loses.

This fact is used in interplanetary travel. It means that the amount of delta-v to reach other planets, over and above that to reach escape velocity can be much less if the delta-v is applied when the rocket is travelling at high speeds, close to the Earth or other planetary surface; whereas waiting until the rocket has slowed at altitude multiplies up the effort required to achieve the desired trajectory.