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Concept
Areal velocity
Summary
In classical mechanics, areal velocity (also called sector velocity or sectorial velocity) is a pseudovector whose length equals the rate of change 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 region swept out by the particle is shaded in green in the figure, bounded by the line segments AB and AC and the curve along which the particle moves. The areal velocity magnitude (i.e., the areal speed) is this region's area divided by the time interval Δt in the limit that Δt becomes vanishingly small. The vector direction is postulated to be normal to the plane containing the position and velocity vectors of the particle, following a convention known as the right hand rule.
Areal velocity is closely related to angular momentum. Any object has an orbital angular momentum about an origin, and this turns out to be, up to a multiplicative scalar constant, equal to the areal velocity of the object about the same origin. A crucial property of angular momentum is that it is conserved under the action of central forces (i.e. forces acting radially toward or away from the origin). Historically, the law of conservation of angular momentum was stated entirely in terms of areal velocity. A special case of this is Kepler's second law, which states that the areal velocity of a planet, with the sun taken as origin, is constant with time. Because the gravitational force acting on a planet is approximately a central force (since the mass of the planet is small in comparison to that of the sun), the angular momentum of the planet (and hence the areal velocity) must remain (approximately) 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.
Official source
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