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[[File:OrbitalEccentricityDemo.svg|thumb|right|The green path in this image is an example of a parabolic trajectory.]]
[[File:Gravity Wells Potential Plus Kinetic Energy - Circle-Ellipse-Parabola-Hyperbola.png|thumb|250px|A parabolic trajectory is depicted in the bottom-left quadrant of this diagram, where the [[gravity well|gravitational potential well]] of the central mass shows potential energy, and the kinetic energy of the parabolic trajectory is shown in red. The height of the kinetic energy decreases asymptotically toward zero as the speed decreases and distance increases according to Kepler's laws.]]
{{about|a class of Kepler orbits|a free body trajectory at constant gravity|Ballistic trajectory}}
In [[astrodynamics]] or [[celestial mechanics]] a '''parabolic trajectory''' is a [[Kepler orbit]] with the [[Orbital eccentricity|eccentricity]] equal to 1. When moving away from the source it is called an '''escape orbit''', otherwise a '''capture orbit'''. It is also sometimes referred to as a '''C<sub>3</sub>&nbsp;=&nbsp;0 orbit''' (see [[Characteristic energy]]).


Under standard assumptions a body traveling along an escape orbit will [[orbital coast|coast]] along a [[Parabola|parabolic]] shaped trajectory to infinity, with velocity relative to the [[central body]] tending to zero, and therefore will never return. Parabolic trajectories are minimum-energy escape trajectories, separating positive-[[characteristic energy|energy]] [[hyperbolic trajectory|hyperbolic trajectories]] from negative-energy [[elliptic orbit]]s.
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==Velocity==
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Under standard assumptions the [[Kinetic energy|orbital velocity]] (<math>v\,</math>) of a body travelling along parabolic trajectory can be computed as:
:<math>v=\sqrt{2\mu\over{r}}</math>
where:
*<math>r\,</math> is the radial distance of orbiting body from [[central body]],
*<math>\mu\,</math> is the [[standard gravitational parameter]].


At any position the orbiting body has the [[escape velocity]] for that position.
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If the body has the escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun.
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This velocity (<math>v\,</math>) is closely related to the [[Kinetic energy|orbital velocity]] of a body in a [[circular orbit]] of the radius equal to the radial position of orbiting body on the parabolic trajectory:
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:<math>v=\sqrt{2}\cdot v_o</math>
 
where:
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*<math>v_o\,</math> is [[Kinetic energy|orbital velocity]] of a body in [[circular orbit]].
 
 
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==Equation of motion==
 
Under standard assumptions, for a body moving along this kind of [[orbit|trajectory]] an [[orbital equation]] becomes:
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:<math>r={{h^2}\over{\mu}}{{1}\over{1+\cos\nu}}</math>
 
where:
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*<math>r\,</math> is radial distance of orbiting body from [[central body]],
 
*<math>h\,</math> is [[specific angular momentum]] of the [[orbiting body]],
</ul>
*<math>\nu\,</math> is a [[true anomaly]] of the orbiting body,
*<math>\mu\,</math> is the [[standard gravitational parameter]].
 
==Energy==
Under standard assumptions, [[specific orbital energy]] (<math>\epsilon\,</math>) of parabolic trajectory is zero, so the [[orbital energy conservation equation]] for this trajectory takes form:
:<math>\epsilon={v^2\over2}-{\mu\over{r}}=0</math>
where:
*<math>v\,</math> is [[Kinetic energy|orbital velocity]] of orbiting body,
*<math>r\,</math> is radial distance of orbiting body from [[central body]],
*<math>\mu\,</math> is the [[standard gravitational parameter]].
 
This is entirely equivalent to the [[characteristic energy]] (square of the speed at infinity) being 0:
 
:<math>C_3 = 0</math>
 
==Barker's equation==
Barker's equation relates the time of flight to the true anomaly of a parabolic trajectory.<ref>{{cite book
| last1 = Bate
| first1 = Roger
| last2 = Mueller
| first2 = Donald
| last3 = White
| first3 = Jerry
| title = Fundamentals of Astrodynamics
| publisher = Dover Publications, Inc., New York
| year = 1971
| ISBN = 0-486-60061-0}} p 188</ref>
 
<math>
t - T = \frac{1}{2}\sqrt{\frac{p^{3}}{\mu}}\left (D + \frac{1}{3}D^{3}\right )
</math>
 
Where:
*D = tan(ν/2), ν is the true anomaly of the orbit
*t is the current time in seconds
*T is the time of periapsis passage in seconds
is the standard gravitational parameter
*p is the [[Conic section#Features|semi-latus rectum]] of the trajectory ( p = h<sup>2</sup>/μ )
 
More generally, the time between any two points on an orbit is
 
<math>
t_{f} - t_{0} = \frac{1}{2}\sqrt{\frac{p^{3}}{\mu}}\left (D_{f} + \frac{1}{3}D_{f}^{3} - D_{0} - \frac{1}{3}D_{0}^{3}\right )
</math>
 
Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit r<sub>p</sub> = p/2:
 
<math>
t - T = \sqrt{\frac{2 r_{p}^{3}}{\mu}}\left (D + \frac{1}{3}D^{3}\right )
</math>
 
Unlike [[Kepler's equation]], which is used to solve for true anomalies in elliptical and hyperbolic trajectories, the true anomaly in Barker's equation can be solved directly for t. If the following substitutions are made<ref>
{{cite book
| last1 = Montenbruck
| first1 = Oliver
| last2 = Pfleger
| first2 = Thomas
| title = Astronomy on the Personal Computer
| publisher = Springer-Verlag Berlin Heidelberg
| year = 2009
| ISBN = 978-3-540-67221-0}} p 64</ref>
 
<math>
A = \frac{3}{2}\sqrt{\frac{\mu}{2r_{p}^{3}}}(t-T)
</math>
 
<math>
B = \sqrt[3]{A + \sqrt{A^{2}+1}}
</math>
 
then
 
<math>
\nu = 2\arctan(B - 1/B)
</math>
 
==Radial parabolic trajectory==
A radial parabolic trajectory is a non-periodic [[Radial_trajectory|trajectory on a straight line]] where the relative velocity of the two objects is always the [[escape velocity]]. There are two cases: the bodies move away from each other or towards each other.
 
There is a rather simple expression for the position as function of time:
 
:<math>r=(4.5\mu t^2)^{1/3}\!\,</math>
 
where
* μ is the [[standard gravitational parameter]]
* <math>t=0\!\,</math> corresponds to the extrapolated time of the fictitious starting or ending at the center of the central body.
 
At any time the average speed from <math>t=0\!\,</math> is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.
 
To have <math>t=0\!\,</math> at the surface, apply a time shift; for the Earth (and any other spherically symmetric body with the same average density) as central body this time shift is 6 minutes and 20 seconds; seven of these periods later the height above the surface is three times the radius, etc.
 
==See also==
* [[Kepler orbit]]
* [[Parabola]]
==References==
{{Reflist}}
 
<!-- there is a navbox, such list was redundant -->
{{orbits}}
 
{{DEFAULTSORT:Parabolic Trajectory}}
 
[[Category:Orbits]]

Revision as of 14:41, 4 March 2014

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Can We Blame This One with Education?Finding people who can easily spell is apparently becoming complicated, according to many employers, plus the CBI (Confederation for British Industry) agrees. Research conducted showed that small employees were sadly with a lack of many employability skills including punctuational. In fact some 44 percent regarding companies are having to invest in literacy and numeracy tuition so their workers can perform basic tasks. Over time of employers complaining in which school leavers weren wise in the simple skills needed for the world of work and thinking that academia had been enough, it seems that somewhere along the line education has skewed off course to forgetting maybe the basics.

Your website might be technologically advanced and you probably have excellent technical skills, but old fashioned skills like spelling are still vital. In fact, lack of it could be wrecking the efficiency of your business, as most of your communication will need place using the written concept. If training your staff is really what needed, then it might need a trade on your part in the beginning, but in the future it will end up making you additional money.

Language Is Changing All the Time It Can't Be That Big some sort of DealIf you are one of those people who is adament we shouldn get so hung up about a misplaced letter, understanding that language is always moving on in any case, then you may be unconvinced by this controversy. It true that language is actually evolving, one look at a webpage of Shakespeare is enough to confirm that, but that isn what this is about. I agree that variants in transliteration should be acceptable, especially in Net World where we converse across continents that have various spellings for words anyway. I personally not going to worry if the site uses "judgement" or "judgment", even so don think bad punctuational should be accepted.

A few years ago, some sort of lecturer at Bucks Fresh University argued that after a Jeremy Scott Uk lot of years of correcting the same errors in his students work, shouldn we let it go and let common errors become variants. Well, by that logic, although stealing is wrong, we should all just leave our houses unlocked as well as let the thieves help them selves! Language will always move on, however that doesn mean we should accept evident cases of bad spelling.

Whatrrrs your opinion? Is getting hung up on undesirable spelling archaic, or is still it an important skill to have? Do you think that bad spelling may possibly really cost you money, or you think customers don really detect? Let us know.

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It easy enough to identify individuals he has displeased. They consist of traditionalists, gay and straight, of any major religion Christian, Jewish and Muslim baffled by his insistence on redefining early institution, central to their morals.

As we reveal today, the actual Church of England, currently tearing itself apart about homosexual priests, has raised deep concerns with Faith Minister Sayeeda Warsi over consequences of your new plans.

Will belief schools have to teach that gay Louis Vuitton Bags and heterosexual marriage are identical? And won priests chance being taken to court to get refusing to marry lesbian and gay couples?

Truly, politicians adventure into deep waters when they meddle with religious affairs. Which is why declare and church have historically been kept separate.

Although the really bitter irony is always that Mr Cameron compromise has prevailed in exasperating supporters associated with gay marriage.

Indeed, nearly everyone is bewildered by the idea of allowing it in some churches, when banning it in the Religious organization of England and the (disestablished) Cathedral in Wales.

If it is good enough with regard to Cheap Timberland Boots Uk Muslims and Catholics (not that they permit it), then why not for Anglicans?

When none of this worries Mr Cameron, he should surely fear the effects on his own party.

Since activists leave in droves, tricked by what they see for a deeply unconservative measure, and UKIP rubs its hands in glee, backbenchers have been in despair.

They look at a entire world in turmoil a terrible debt crisis, family earnings squeezed, the eurozone near fail, tensions rising in Syria in addition to Korea and they see the Conventional Party staking its future with a matter that is, frankly, immaterial to most people.

With perceptions changing, the Prime Minister meets your needs to think most voters wish homosexuals very well. But he is wrong to believe this is his IV moment as important to him as being the rejection of nationalisation was to Fresh Labour.

There are much more crucial issues to engage his sizeable talents.

Getting back to workThough economists are generally mystified by the trend, this papers draws huge Mbt Shoes Uk cheer through the biggest quarterly fall throughout unemployment for more than a decade.

Certainly, George Osborne is right to trust confidential firms ability to create careers at double the rate with public sector lay offs.

Should this be what can be achieved during a credit squeeze, imagine how fast a economy could recover in the event the banks faced their liability to lend.

A cautionary taleThe Send congratulates the Daily Telegraph for its courage in creating a story about the misuse with expenses by the very reverend charged with deciding how the Touch is to be regulated.

In phrases whose meaning was all too clear, a Ralph Lauren Outlet Uk special adviser reported she wanted to up Betty Miller involvement with click regulation before the paper discovered the Media Secretary had said 90,000 for a household where her parents likewise lived. Later, it is professed, Number 10 press policeman made a similar veiled threat.

This paper believes Mrs Miller can be a well intentioned and decent, in the event inexperienced minister, who was drastically wrong to complain that the report had spoken to her daddy.

But could any cameo more graphically show the danger of giving politicians your say in controlling newspapers?

In truth, this illuminating story totally vindicates Mr Cameron principled stand against statutory regulation of the Press.

I believe that more to the point is what will happen when the world ends about the 21st of December. I'd like to point out that discrimination happens at all levels of society and it is a natural event due to the diversity of mankind, however while nobody knows where Our god is or what the (my) name is, it ought to be understood that nobody can get away his discrimination. When the prophet Ezekiel found the four living creatures, each individual with the face of a man on the one hand, and a lion ox and bald eagle on the others, he failed to fully understand the meaning of it. An individual oh sons of Our god, can you fly across the world with out understanding? Can you roar plus catch your pray without understanding? Can you till an area without understanding?

It should hence be clear that Goodness alone adds and subtracts, and there's no retreat from his being familiar with.

Mark Joslin 2012 ( I made this all up and therefore ought not to be taken as legend)

Why don't we turn the argument, what right has any bigot to tell a church who wants to got married a gay couple that, no, that's not allowed, that is certainly infringing my rights.

What an utter load of rabble. Next the discussion will be that not reintroducing the suspend on homosexuality infringes the bigots appropriate top put people he or she is bigoted towards in jail.

Stop this particular charade before the hand of history precipitates hard on your shoulder to tell you how you'll be remembered. Together with the 19th Baron Willoughby de Pennyless, I'd judge.