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Areal velocity is the rate at which area is swept by the position vector of a point which moves along a curve. Areal velocity is the magnitude of the areal velocity vector, which is parallel (but not necessarily proportional in magnitude) to the angular velocity vector.
Areal velocity depends on a reference point: the origin of the coordinate system of the position vector, which is a function of time.
Figure 1. Areal velocity is the area (shown in yellow) swept per unit time by a particle moving along a curve (shown in blue).
Figure 1 shows a regular curve in blue. At time t a moving particle is located at point B, and at time t + Δt the same particle has moved to point C.
The area swept (or transcribed) during time period Δt by the particle is nearly equal to the area of triangle ABC. As Δt approaches zero this near equality becomes exact as a limit.
Meanwhile, vectors AB and AC add up by the parallelogram method to vector AD, so that point D is the fourth corner of parallelogram ABDC shown in Figure 1.
The area of triangle ABC (in yellow) 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, so that
- <math> \hbox{area}(ABDC) = \mathbf{r}(t) \times \mathbf{r}(t+\Delta t), <math>
- <math> \hbox{area}(ABC) = {\mathbf{r}(t) \times \mathbf{r}(t + \Delta t) \over 2} = \Delta \vec A_r. <math>
The areal velocity vector is
- <math> \vec\mathbf{\omega}_{A_r} = \lim_{\Delta t \rightarrow 0} {\Delta \vec A_r \over \Delta t} = \lim_{\Delta t \rightarrow 0} {\mathbf{r}(t) \times \mathbf{r}(t + \Delta t) \over 2 \Delta t}<math>
- <math> = \lim_{\Delta t \rightarrow 0} {\mathbf{r}(t) \times [ \mathbf{r}(t) + \mathbf{r'}(t) \Delta t ] \over 2 \Delta t} <math>
- <math> = \lim_{\Delta t \rightarrow 0} {\mathbf{r}(t) \times \mathbf{r'}(t) \over 2} \left( {\Delta t \over \Delta t} \right) <math>
- <math> \vec\omega_{A_r} = {\mathbf{r}(t) \times \mathbf{r'}(t) \over 2}. <math>
But r′(t) is the linear velocity vector v(t), so that
- <math> \vec\omega_{A_r} = {\mathbf{r} \times \mathbf{v} \over 2}. <math>
Kepler's second law of planetary motion is a statement of conservation of areal velocity of the orbiting planet with respect to the Sun.
Notice that twice the areal velocity times mass equals angular momentum, just as linear velocity times mass is linear momentum, i.e.
- linear velocity : linear momentum :: (double) areal velocity : angular momentum.
See also
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