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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.
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Classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility). The "classical" in "classical mechanics" does not refer classical antiquity, as it might in, say, classical architecture.
Velocity
Velocity is the speed and the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity: both magnitude and direction are needed to define it. The scalar absolute value (magnitude) of velocity is called , being a coherent derived unit whose quantity is measured in the SI (metric system) as metres per second (m/s or m⋅s−1). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector.
Angular momentum
In physics, angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum.