UGM-133 Trident II
The UGM-133A Trident II, or Trident D5 is a submarine-launched ballistic missile, built by Lockheed Martin Space Systems in Sunnyvale and deployed with the American and British navies. It was first deployed in March 1990, remains in service; the Trident II Strategic Weapons System is an improved SLBM with greater accuracy and range than the earlier Trident C-4. It is a key element of the U. S. strategic nuclear triad and strengthens U. S. strategic deterrence. The Trident II is considered to be a durable sea-based system capable of engaging many targets, it enhances the U. S. position in strategic arms negotiation with performance and payload flexibility that can accommodate active treaty initiatives. The Trident II's increased payload allows nuclear deterrence to be accomplished with fewer submarines, its high accuracy – approaching that of land-based missiles – enables it to be used as a first strike weapon. Trident II missiles are carried by 14 US Ohio and four British Vanguard-class submarines, with 24 missiles on each Ohio class and 16 missiles on each Vanguard class.
There have been over 161 successful test flights of the D5 missile since design completion in 1989, the most recent being from USS Nebraska in March 2018. There have been fewer than 10 test flights that were failures, the most recent being from HMS Vengeance off the coast of Florida in June 2016; the D5 is the sixth in a series of missile generations deployed since the sea-based deterrent program began 60 years ago. The Trident D5LE version will remain in service until 2042; the Trident II was designated to be the latest longer-range missile, performing greater than its predecessor in terms of range and payload capacity. In 1972, the US Navy projected an initial operating capability date for the Trident II in 1984; the US Navy continued to advance the IOC date to 1982. On 18 October 1973, a Trident program review was administered. On 14 March 1974, the US Deputy Secretary of Defense disseminated two requirements for the Trident program; the first was an accuracy improvement for the Trident C-4.
The second requirement asked for an alternative to the C-4, or a new Trident II missile with a larger first stage motor than the C-4. Studies were conducted to determine whether the more-expensive Trident II could be constructed similar to the US Air Force's MX ICBM; this was done to decrease budget costs. It was established that the Trident II would be 83 inches in diameter and 44 ft in length in order to maintain performance with the existing MX ICBM. Modifications to the guidance system, electronics hardening, external protective coatings were incorporated into the design. While this satisfied the US Naval study requirements, it did not accommodate the US Air Force payload requirements. Propulsion stages were proposed to be used between the first stage and second stage motors making the Trident II a longer three-stage missile than the C-4. Studies were delayed in 1978 when Congress only approved $5 million of the suggested $15 million for the Naval/Air Force program studies. By December 1978, the US Navy and Air Force own studies agreed with each other that the savings made by a similar missile structure would not be effective.
It was determined that the US Navy and Air Force maintain and be responsible for their own unique weapon systems. The US Navy continued with their own weapon design of the Trident II. In March 1980, US Secretary of Defense Harold Brown proposed an increased level of funding for the submarine-launched ballistic missile modernization. Emphasis was strained for the need of increased accuracy; the House Armed Services Committee recommended no funding, while the Senate Armed Services Committee recommended full funding of $97 million. The SASC asked for a plan which incorporates "the fullest possible competition... should consider competing among contractors for each major component, including the integrated missile." $65 million was awarded for the submarine-launched ballistic missile modernization. On 2 October 1981, President Reagan called for the modernization of the strategic forces; the Defense Department directed the Navy to fund all development of the Trident II D5 missile with a December 1989 IOC.
All research and development effort would be directed toward "a new development, advanced technology, high accuracy Trident II D5 system." In December 1982, Deputy SECDEF Frank Carlucci advised Secretary of the Navy Caspar Weinberger to include funding for a new RV/warhead combination for Trident II. The reentry vehicle was to be designated as the Mk 5, to have an increased yield than the Mk 4; the development contract for Trident II was issued in October 1983. On 28 December 1983, the Deputy SECDEF authorized the Navy to proceed with Full Scale Engineering Development of the Trident II D5; the first Trident II launch occurred on 15 January 1987, the first submarine launch was attempted by USS Tennessee, the first D-5 ship of the Ohio class, on 21 March 1989 off the coast Cape Canaveral, Florida. The launch attempt failed four seconds into the flight because the plume of water following the missile rose to greater height than expected, resulting in water being in the nozzle when the motor ignited.
Once the problem was understood simple changes were made, but the problem delayed the IOC of Trident II until March 1990. IOC for SWFPAC completed on schedule in 2001 which allowed Trident II SSBN to be supported in the Pacific theater. In 1980, the United Kingdom adopted the missile as part of its Trident nuclear program; the Trident II is a three-stage rocket, each stage containing a so
Sagarika known by the code names K-15 or B-05, is a nuclear-capable submarine-launched ballistic missile with a range of 750 kilometres. It belongs to the K Missile family and forms a part of India's nuclear triad, will provide retaliatory nuclear strike capability. Development of the K-15 missile started in the late 1990s with the goal of building a submarine-launched ballistic missile for use with the Indian Navy nuclear-powered Arihant-class submarines, it was developed at the Defence Research and Development Organisation’s missile complex in Hyderabad. The development of the underwater missile launcher, known as Project 420, was completed in 2001 and handed over to the Indian Navy for trials; the missile launcher is developed at Hazira in Gujarat. The Sagarika missile began integration with India's nuclear-powered Arihant class submarine that began harbor trials on 26 July 2009. By 2008, the missile was test fired seven times, tested to its full range up to four times; the tests of 26 February 2008, were conducted from a submerged pontoon 50 metres beneath the surface off the coast of Visakhapatnam.
A land-based version of the K-15 Sagarika was test-fired on 12 November 2008. A full range test of the missile was done on 11 March 2012; the twelfth and final development trial of the missiles was conducted on 27 January 2013. According to DRDO Director General V. K. Saraswat, the missile was again tested for its full range of 700 km and met all its objectives with a single digit impact-accuracy; the test will be followed by integration of the missile with INS Arihant. On 25 November 2015, a dummy or unarmed K-15 Sagarika missile was test fired from INS Arihant. K Missile family Arihant-class submarine K-4 CSIS Missile Threat - Sagarika/Shaurya Video of the 12th test of Sagarika, on 27 January 2013
Submarine-launched ballistic missile
A submarine-launched ballistic missile is a ballistic missile capable of being launched from submarines. Modern variants deliver multiple independently targetable reentry vehicles each of which carries a nuclear warhead and allows a single launched missile to strike several targets. Submarine-launched ballistic missiles operate in a different way from submarine-launched cruise missiles. Modern submarine-launched ballistic missiles are related to intercontinental ballistic missiles (with ranges of over 5,500 kilometres, in many cases SLBMs and ICBMs may be part of the same family of weapons; the first practical design of a submarine-based launch platform was developed by the Germans near the end of World War II involving a launch tube which contained a V-2 ballistic missile variant and was towed behind a submarine, known by the code-name Prüfstand XII. The war ended before it could be tested, but the engineers who had worked on it went on to work for the United States and for the Soviet Union on their SLBM programs.
These and other early SLBM systems required vessels to be surfaced when they fired missiles, but launch systems were adapted to allow underwater launching in the 1950-1960s. A converted Project 611 submarine launched the world's first SLBM, an R-11FM on 16 September 1955. Five additional Project V611 and AV611 Submarines became the world's first operational ballistic missile submarines with two R-11FM missiles each, entering service in 1956–57; the United States Navy worked on a sea-based variant of the US Army Jupiter intermediate-range ballistic missile, projecting four of the large, liquid-fueled missiles per submarine. Rear Admiral W. F. "Red" Raborn headed a Special Project Office to develop Jupiter for the Navy, beginning in late 1955. However, at the Project Nobska submarine warfare conference in 1956, physicist Edward Teller stated that a physically small one-megaton warhead could be produced for the small, solid-fueled Polaris missile, this prompted the Navy to leave the Jupiter program in December of that year.
Soon Chief of Naval Operations Admiral Arleigh Burke concentrated all Navy strategic research on Polaris, still under Admiral Raborn's Special Project Office. All US SLBMs have been solid-fueled while all Soviet and Russian SLBMs have been liquid-fueled except for the Russian RSM-56 Bulava, which entered service in 2014; the world's first operational nuclear-powered ballistic missile submarine was USS George Washington with 16 Polaris A-1 missiles, which entered service in December 1959 and conducted the first SSBN deterrent patrol November 1960 – January 1961. George Washington conducted the first successful submerged SLBM launch with a Polaris A-1 on 20 July 1960. Forty days the Soviet Union made its first successful underwater launch of a submarine ballistic missile in the White Sea, on 10 September 1960 from the same converted Project 611 submarine that first launched the R-11FM; the Soviets were only a year behind the US with their first SSBN, the ill-fated K-19 of Project 658, commissioned in November 1960.
However, the Hotel class carried only three R-13 missiles each and had to surface and raise the missile to launch. Submerged launch was not an operational capability for the Soviets until 1963, when the R-21 missile was first backfitted to Project 658 and Project 629 submarines; the Soviet Union was able to beat the U. S. in launching and testing the first SLBM with a live nuclear warhead, an R-13 that detonated in the Novaya Zemlya Test Range in the Arctic Ocean, doing so on 20 October 1961, just ten days before the gigantic 50 Mt Tsar Bomba's detonation in the same general area. The United States conducted a similar test in the Pacific Ocean on 6 May 1962, with a Polaris A-2 launched from USS Ethan Allen as part of the nuclear test series Operation Dominic; the first Soviet SSBN with 16 missiles was the Project 667A, which first entered service in 1967 with 32 boats completed by 1974. By the time the first Yankee was commissioned the US had built 41 SSBNs, nicknamed the "41 for Freedom"; the short range of the early SLBMs dictated deployment locations.
By the late 1960s the Polaris A-3 was deployed on all US SSBNs with a range of 4,600 kilometres, a great improvement on the 1,900 kilometres range of Polaris A-1. The A-3 had three warheads that landed in a pattern around a single target; the Yankee class was equipped with the R-27 Zyb missile with a range of 2,400 kilometres. The US was much more fortunate in its basing arrangements than the Soviets. Thanks to NATO and the US possession of Guam, US SSBNs were permanently forward deployed at Advanced Refit Sites in Holy Loch, Rota and Guam by the middle 1960s, resulting in short transit times to patrol areas near the Soviet Union; the SSBN facilities at the Advanced Refit Sites were austere, with only a submarine tender and floating dry dock. Converted merchant ships designated T-AKs were provided to ferry supplies to the sites. With two rotating crews per boat, about one-third of the total US force could be in a patrol area at any time; the Soviet bases, in the Murmansk area for the Atlantic and the Petropavlovsk-Kamchatsky area for the Pacific, required their SSBNs to make a long transit to their mid-ocean patrol areas to hold the continental United States at risk.
This resulted in only a small percentage of the Soviet force occupying patrol areas at any time, an
Nazi Germany is the common English name for Germany between 1933 and 1945, when Adolf Hitler and his Nazi Party controlled the country through a dictatorship. Under Hitler's rule, Germany was transformed into a totalitarian state that controlled nearly all aspects of life via the Gleichschaltung legal process; the official name of the state was Deutsches Reich until 1943 and Großdeutsches Reich from 1943 to 1945. Nazi Germany is known as the Third Reich, meaning "Third Realm" or "Third Empire", the first two being the Holy Roman Empire and the German Empire; the Nazi regime ended. Hitler was appointed Chancellor of Germany by the President of the Weimar Republic, Paul von Hindenburg, on 30 January 1933; the NSDAP began to eliminate all political opposition and consolidate its power. Hindenburg died on 2 August 1934 and Hitler became dictator of Germany by merging the offices and powers of the Chancellery and Presidency. A national referendum held 19 August 1934 confirmed Hitler as sole Führer of Germany.
All power was centralised in Hitler's person and his word became the highest law. The government was not a coordinated, co-operating body, but a collection of factions struggling for power and Hitler's favour. In the midst of the Great Depression, the Nazis restored economic stability and ended mass unemployment using heavy military spending and a mixed economy. Extensive public works were undertaken, including the construction of Autobahnen; the return to economic stability boosted the regime's popularity. Racism antisemitism, was a central feature of the regime; the Germanic peoples were considered by the Nazis to be the master race, the purest branch of the Aryan race. Discrimination and persecution against Jews and Romani people began in earnest after the seizure of power; the first concentration camps were established in March 1933. Jews and others deemed undesirable were imprisoned, liberals and communists were killed, imprisoned, or exiled. Christian churches and citizens that opposed Hitler's rule were oppressed, many leaders imprisoned.
Education focused on racial biology, population policy, fitness for military service. Career and educational opportunities for women were curtailed. Recreation and tourism were organised via the Strength Through Joy program, the 1936 Summer Olympics showcased Germany on the international stage. Propaganda Minister Joseph Goebbels made effective use of film, mass rallies, Hitler's hypnotic oratory to influence public opinion; the government controlled artistic expression, promoting specific art forms and banning or discouraging others. The Nazi regime dominated neighbours through military threats in the years leading up to war. Nazi Germany made aggressive territorial demands, threatening war if these were not met, it seized Austria and Czechoslovakia in 1938 and 1939. Germany signed a non-aggression pact with the USSR, invaded Poland on 1 September 1939, launching World War II in Europe. By early 1941, Germany controlled much of Europe. Reichskommissariats took control of conquered areas and a German administration was established in the remainder of Poland.
Germany exploited labour of both its occupied territories and its allies. In the Holocaust, millions of Jews and other peoples deemed undesirable by the state were imprisoned, murdered in Nazi concentration camps and extermination camps, or shot. While the German invasion of the Soviet Union in 1941 was successful, the Soviet resurgence and entry of the US into the war meant the Wehrmacht lost the initiative on the Eastern Front in 1943 and by late 1944 had been pushed back to the pre-1939 border. Large-scale aerial bombing of Germany escalated in 1944 and the Axis powers were driven back in Eastern and Southern Europe. After the Allied invasion of France, Germany was conquered by the Soviet Union from the east and the other Allies from the west, capitulated in May 1945. Hitler's refusal to admit defeat led to massive destruction of German infrastructure and additional war-related deaths in the closing months of the war; the victorious Allies initiated a policy of denazification and put many of the surviving Nazi leadership on trial for war crimes at the Nuremberg trials.
The official name of the state was Deutsches Reich from 1933 to 1943 and Großdeutsches Reich from 1943 to 1945, while common English terms are "Nazi Germany" and "Third Reich". The latter, adopted by Nazi propaganda as Drittes Reich, was first used in Das Dritte Reich, a 1923 book by Arthur Moeller van den Bruck; the book counted the Holy Roman Empire as the German Empire as the second. Germany was known as the Weimar Republic during the years 1919 to 1933, it was a republic with a semi-presidential system. The Weimar Republic faced numerous problems, including hyperinflation, political extremism, contentious relationships with the Allied victors of World War I, a series of failed attempts at coalition government by divided political parties. Severe setbacks to the German economy began after World War I ended because of reparations payments required under the 1919 Treaty of Versailles; the government printed money to make the payments and to repay the country's war debt, but the resulting hyperinflation led to inflated prices for consumer goods, economic chaos, food riots.
When the government defaulted on their reparations payments in January 1923, French troops occupied German industrial areas along the Ruhr and widespread civil unrest followed. The National Socialist German Workers' Party (National
History of rockets
The first rockets were used as propulsion systems for arrows, may have appeared as early as the 10th century in Song dynasty China. However more solid documentary evidence does not appear until the 13th century; the technology spread across Eurasia in the wake of the Mongol invasions of the mid-13th century. Usage of rockets as weapons before modern rocketry is attested in China, Indian subcontinent, Europe. One of the first recorded rocket launchers is the "wasp nest" fire arrow launcher produced by the Ming dynasty in 1380. In Europe rockets were used in the same year at the Battle of Chioggia; the Joseon kingdom of Korea used a type of mobile multiple rocket launcher known as the "Munjong Hwacha" by 1451. Iron-cased rockets, known as Mysorean rockets, were developed in Kingdom of Mysore by the mid 18th century in India, were copied by the British; the models and improvements were known as the Congreve rocket and used in the Napoleonic Wars. The dating of the invention of the first rocket, otherwise known as the gunpowder propelled fire arrow, is disputed.
The History of Song attributes the invention to two different people at different times, Feng Zhisheng in 969 and Tang Fu in 1000. However Joseph Needham argues that rockets could not have existed before the 12th century, since the gunpowder formulas listed in the Wujing Zongyao are not suitable as rocket propellant. Rockets may have been used as early as 1232, when reports appeared describing fire arrows and'iron pots' that could be heard for 5 leagues when they exploded upon impact, causing devastation for a radius of 600 meters due to shrapnel. Rockets are recorded to have been used by the Song navy in a military exercise dated to 1245. Internal-combustion rocket propulsion is mentioned in a reference to 1264, recording that the'ground-rat,' a type of firework, had frightened the Empress-Mother Gongsheng at a feast held in her honor by her son the Emperor Lizong. Subsequently, rockets are included in the military treatise Huolongjing known as the Fire Drake Manual, written by the Chinese artillery officer Jiao Yu in the mid-14th century.
This text mentions the first known multistage rocket, the'fire-dragon issuing from the water', thought to have been used by the Chinese navy. Rocket launchers known as "wasp nests" were ordered by the Ming army in 1380; the American historian Frank H. Winter proposed in The Proceedings of the Twentieth and Twenty-First History Symposia of the International Academy of Astronautics that southern China and the Laotian community rocket festivals might have been key in the subsequent spread of rocketry in the Orient; the Chinese fire arrow was adopted by the Mongols in northern China, who employed Chinese rocketry experts as mercenaries in the Mongol army. Rockets are thought to have spread via the Mongol invasions to other areas of Eurasia in the mid 13th century. Rocket-like weapons are reported to have been used at the Battle of Mohi in the year 1241. Between 1270 and 1280, Hasan al-Rammah wrote his al-furusiyyah wa al-manasib al-harbiyya, which included 107 gunpowder recipes, 22 of which are for rockets.
According to Ahmad Y Hassan, al-Rammah's recipes were more explosive than rockets used in China at the time. The terminology used by al-Rammah indicates a Chinese origin for the gunpowder weapons he wrote about, such as rockets and fire lances. Ibn al-Baitar, an Arab from Spain who had immigrated to Egypt, described saltpeter as "snow of China". Al-Baytar died in 1248; the earlier Arab historians called saltpeter "Chinese snow" and " Chinese salt." The Arabs used the name "Chinese arrows" to refer to rockets. The Arabs called fireworks "Chinese flowers". While saltpeter was called "Chinese Snow" by Arabs, it was called "Chinese salt" by the Iranians, or "salt from the Chinese marshes". In 1300 Mongol mercenaries in India are recorded to have used hand held rockets. By the mid-14th century Indians were using rockets in warfare; the Korean kingdom of Joseon started producing gunpowder in 1374 and was producing cannons and rockets by 1377. However the multiple rocket launching carts known as the "Munjong hwacha" did not appear until 1451.
In Europe, Roger Bacon mentions gunpowder in his Opus Majus of 1267. However rockets do not feature in European warfare until the 1380 Battle of Chioggia. Konrad Kyeser described rockets in his famous military treatise Bellifortis around 1405. Jean Froissart had the idea of launching rockets through tubes, so that they could make more accurate flights. Froissart's idea is a forerunner of the modern bazooka. According to the 18th-century historian Ludovico Antonio Muratori, rockets were used in the war between the Republics of Genoa and Venice at Chioggia in 1380, it is uncertain whether Muratori was correct in his interpretation, as the reference might have been to bombard, but Muratori is the source for the widespread claim that the earliest recorded European use of rocket artillery dates to 1380. Konrad Kyeser described rockets in his famous military treatise Bellifortis around 1405. Kyeser describes three types of rockets, free flying and captive. Joanes de Fontana in Bellicorum instrumentorum liber described flying rockets in the shape of doves, running rockets in the shape of hares, a large car driven by three rockets, as well as a large rocket torpedo with the head of a sea monster.
In the mid-16th century, Conrad Haas wrote a book that described rocket technology that combined fireworks and weapons technologies. This manuscript was discovered in the Sibiu public records, his work d
A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission or from a combination of fission and fusion reactions. Both bomb types release large quantities of energy from small amounts of matter; the first test of a fission bomb released an amount of energy equal to 20,000 tons of TNT. The first thermonuclear bomb test released energy equal to 10 million tons of TNT. A thermonuclear weapon weighing little more than 2,400 pounds can release energy equal to more than 1.2 million tons of TNT. A nuclear device no larger than traditional bombs can devastate an entire city by blast and radiation. Since they are weapons of mass destruction, the proliferation of nuclear weapons is a focus of international relations policy. Nuclear weapons have been used twice in war, both times by the United States against Japan near the end of World War II. On August 6, 1945, the U. S. Army Air Forces detonated a uranium gun-type fission bomb nicknamed "Little Boy" over the Japanese city of Hiroshima.
S. Army Air Forces detonated a plutonium implosion-type fission bomb nicknamed "Fat Man" over the Japanese city of Nagasaki; these bombings caused injuries that resulted in the deaths of 200,000 civilians and military personnel. The ethics of these bombings and their role in Japan's surrender are subjects of debate. Since the atomic bombings of Hiroshima and Nagasaki, nuclear weapons have been detonated over two thousand times for testing and demonstration. Only a few nations are suspected of seeking them; the only countries known to have detonated nuclear weapons—and acknowledge possessing them—are the United States, the Soviet Union, the United Kingdom, China, India and North Korea. Israel is believed to possess nuclear weapons, though, in a policy of deliberate ambiguity, it does not acknowledge having them. Germany, Turkey and the Netherlands are nuclear weapons sharing states. South Africa is the only country to have independently developed and renounced and dismantled its nuclear weapons.
The Treaty on the Non-Proliferation of Nuclear Weapons aims to reduce the spread of nuclear weapons, but its effectiveness has been questioned, political tensions remained high in the 1970s and 1980s. Modernisation of weapons continues to this day. There are two basic types of nuclear weapons: those that derive the majority of their energy from nuclear fission reactions alone, those that use fission reactions to begin nuclear fusion reactions that produce a large amount of the total energy output. All existing nuclear weapons derive some of their explosive energy from nuclear fission reactions. Weapons whose explosive output is from fission reactions are referred to as atomic bombs or atom bombs; this has long been noted as something of a misnomer, as their energy comes from the nucleus of the atom, just as it does with fusion weapons. In fission weapons, a mass of fissile material is forced into supercriticality—allowing an exponential growth of nuclear chain reactions—either by shooting one piece of sub-critical material into another or by compression of a sub-critical sphere or cylinder of fissile material using chemically-fueled explosive lenses.
The latter approach, the "implosion" method, is more sophisticated than the former. A major challenge in all nuclear weapon designs is to ensure that a significant fraction of the fuel is consumed before the weapon destroys itself; the amount of energy released by fission bombs can range from the equivalent of just under a ton to upwards of 500,000 tons of TNT. All fission reactions generate the remains of the split atomic nuclei. Many fission products are either radioactive or moderately radioactive, as such, they are a serious form of radioactive contamination. Fission products are the principal radioactive component of nuclear fallout. Another source of radioactivity is the burst of free neutrons produced by the weapon; when they collide with other nuclei in surrounding material, the neutrons transmute those nuclei into other isotopes, altering their stability and making them radioactive. The most used fissile materials for nuclear weapons applications have been uranium-235 and plutonium-239.
Less used has been uranium-233. Neptunium-237 and some isotopes of americium may be usable for nuclear explosives as well, but it is not clear that this has been implemented, their plausible use in nuclear weapons is a matter of dispute; the other basic type of nuclear weapon produces a large proportion of its energy in nuclear fusion reactions. Such fusion weapons are referred to as thermonuclear weapons or more colloquially as hydrogen bombs, as they rely on fusion reactions between isotopes of hydrogen. All such weapons derive a significant portion of their energy from fission reactions used to "trigger" fusion reactions, fusion reactions can themselves trigger additional fission reactions. Only six countries—United States, United Kingdom, China and India—have conducted thermonuclear weapon tests. North Korea claims to have tested a fusion weapon as of January 2016. Thermonuclear weapons a
Projectile motion is a form of motion experienced by an object or particle, thrown near the Earth's surface and moves along a curved path under the action of gravity only. This curved path was shown by Galileo to be a parabola; the study of such motions is called ballistics, such a trajectory is a ballistic trajectory. The only force of significance that acts on the object is gravity, which acts downward, thus imparting to the object a downward acceleration; because of the object's inertia, no external horizontal force is needed to maintain the horizontal velocity component of the object. Taking other forces into account, such as friction from aerodynamic drag or internal propulsion such as in a rocket, requires additional analysis. A ballistic missile is a missile only guided during the brief initial powered phase of flight, whose subsequent course is governed by the laws of classical mechanics. Ballistics is the science of mechanics that deals with the flight and effects of projectiles bullets, unguided bombs, rockets, or the like.
The elementary equations of ballistics neglect nearly every factor except for initial velocity and an assumed constant gravitational acceleration. Practical solutions of a ballistics problem require considerations of air resistance, cross winds, target motion, varying acceleration due to gravity, in such problems as launching a rocket from one point on the Earth to another, the rotation of the Earth. Detailed mathematical solutions of practical problems do not have closed-form solutions, therefore require numerical methods to address. Let the projectile be launched with an initial velocity v ≡ v 0, which can be expressed as the sum of horizontal and vertical components as follows: v 0 = v 0 x i + v 0 y j; the components v 0 x and v 0 y can be found if the initial launch angle, θ, is known: v 0 x = v 0 cos θ, v 0 y = v 0 sin θ. In projectile motion, the horizontal motion and the vertical motion are independent of each other; this is the principle of compound motion established by Galileo in 1638, used by him to prove the parabolic form of projectile motion.
A ballistic trajectory is a parabola with homogeneous acceleration, such as in a space ship with constant acceleration in absence of other forces. On Earth the acceleration changes magnitude with direction with latitude/longitude; this causes an elliptic trajectory, close to a parabola on a small scale. However, if an object was thrown and the Earth was replaced with a black hole of equal mass, it would become obvious that the ballistic trajectory is part of an elliptic orbit around that black hole, not a parabola that extends to infinity. At higher speeds the trajectory can be circular, parabolic or hyperbolic. In this article a homogeneous acceleration is assumed. Since there is only acceleration in the vertical direction, the velocity in the horizontal direction is constant, being equal to v 0 cos θ; the vertical motion of the projectile is the motion of a particle during its free fall. Here the acceleration is constant, being equal to g; the components of the acceleration are- a x = 0, a y = − g.
The horizontal component of the velocity of the object remains unchanged throughout the motion. The vertical component of the velocity changes linearly, because the acceleration due to gravity is constant; the accelerations in the x and y directions can be integrated to solve for the components of velocity at any time t, as follows: v x = v 0 cos , v y = v 0 sin − g t. The magnitude of the velocity: v = v x 2 + v y 2. At any time t, the projectile's horizontal and vertical displacement are: x = v 0 t cos , y = v 0 t sin −