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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 .
Thus, the areal velocity is constant for a particle acted upon by any type of central force; this is Kepler's second law. [13] Conversely, if the motion under a conservative force F is planar and has constant areal velocity for all initial conditions of the radius r and velocity v, then the azimuthal acceleration a φ is always zero.
As it does so, the object's motion will be described by two vectors: a translation vector, and a rotation vector ω, which is an areal velocity vector: the Darboux vector. Note that this rotation is kinematic , rather than physical, because usually when a rigid object moves freely in space its rotation is independent of its translation.
This constant areal velocity can be calculated as follows. At the apapsis and periapsis, the positions of closest and furthest distance from the attracting center, the velocity and radius vectors are perpendicular; therefore, the angular momentum L 1 per mass m of the particle (written as h 1) can be related to the rate of sweeping out areas
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The velocity vector of the particle is ... The energy conservation equation for the particle is then obtained if f(r) ... Areal velocity; Central force;
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