Egyptian numerals

Areal velocity is the area (shown in green) swept out per unit time by a particle moving along a curve (shown in blue).

In classical mechanics, areal velocity (also called sector velocity or sectorial velocity) is the rate at which area is swept out by a particle as it moves along a curve. In the adjoining figure, suppose that a particle moves along the blue curve. At a certain time t, the particle is located at point B, and a short while later, at time t + Δt, the particle has moved to point C. The area swept out by the article is the green area in the figure, bounded by the line segments AB and AC and the curve along which the particle moves. The areal velocity equals this area divided by the time interval Δt in the limit that Δt becomes vanishingly small.

Illustration of Kepler's second law. The planet moves faster near the Sun, so the same area is swept out in a given time as at larger distances, where the planet moves more slowly.

The concept of areal velocity is closely linked historically with the concept of angular momentum. Kepler's second law states that the areal velocity of a planet, with the sun taken as origin, is constant. Isaac Newton was the first scientist to recognize the dynamical significance of Kepler's second law. With the aid of his laws of motion, he proved in 1684 that any planet that is attracted to a fixed center sweeps out equal areas in equal intervals of time. By the middle of the 18th century, the principle of angular momentum was discovered gradually by Daniel Bernoulli and Leonhard Euler and Patrick d'Arcy; d'Arcy's version of the principle was phrased in terms of swept area. For this reason, the principle of angular momentum was often referred to in the older literature in mechanics as "the principle of areas." Since the concept of angular momentum includes more than just geometry, the designation "principle of areas" has been dropped in modern works.

Connection with angular momentum

In the situation of the first figure, the area swept out during time period Δt by the particle is approximately equal to the area of triangle ABC. As Δt approaches zero this near-equality becomes exact as a limit.

Let the point D be the fourth corner of parallelogram ABDC shown in the figure, so that the vectors AB and AC add up by the parallelogram rule to vector AD. Then the area of triangle ABC is half the area of parallelogram ABDC, and the area of ABDC is equal to the magnitude of the cross product of vectors AB and AC. This area can also be viewed as a vector with this magnitude, pointing in a direction perpendicular to the parallellogram; this vector is the cross product itself:

${\text{vector area of parallellogram }}ABCD={\vec {r}}(t)\times {\vec {r}}(t+\Delta t).$

Hence

${\text{vector area of triangle }}ABC={\frac {{\vec {r}}(t)\times {\vec {r}}(t+\Delta t)}{2}}.$

The areal velocity is this area divided by Δt in the limit that Δt becomes vanishingly small:

${\begin{aligned}{\text{areal velocity}}&=\lim _{\Delta t\rightarrow 0}{\frac {{\vec {r}}(t)\times {\vec {r}}(t+\Delta t)}{2\Delta t}}\\&=\lim _{\Delta t\rightarrow 0}{\frac {{\vec {r}}(t)\times {\bigl (}{\vec {r}}(t)+{\vec {r}}\,'(t)\Delta t{\bigr )}}{2\Delta t}}\\&=\lim _{\Delta t\rightarrow 0}{\frac {{\vec {r}}(t)\times {\vec {r}}\,'(t)}{2}}\left({\Delta t \over \Delta t}\right)\\&={\frac {{\vec {r}}(t)\times {\vec {r}}\,'(t)}{2}}.\end{aligned}}$

But, ${\vec {r}}\,'(t)$ is the velocity vector ${\vec {v}}(t)$ of the moving particle, so that

${\frac {d{\vec {A}}}{dt}}={\frac {{\vec {r}}\times {\vec {v}}}{2}}.$

On the other hand, the angular momentum of the particle is

${\vec {L}}={\vec {r}}\times m{\vec {v}},$

and hence the angular momentum equals 2m times the areal velocity.

Conservation of areal velocity is a general property of central force motion.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

See also

• Angular momentum
• Specific angular momentum
• Elliptical coordinates