The fact that $\mathbf { r } ( t ) \times \mathbf { v } ( t )$ is constant implies that

A)The speed of a planet around the sun is a constant.
B)The motion of the planet lies in a plane.
C)The speed of the sun is a constant.
D)The force between the planet and the sun is a constant.
E)The position of the planet is unrelated to the sun.

Question

The fact that $\mathbf { r } ( t ) \times \mathbf { v } ( t )$ is constant implies that

A)The speed of a planet around the sun is a constant.
B)The motion of the planet lies in a plane.
C)The speed of the sun is a constant.
D)The force between the planet and the sun is a constant.
E)The position of the planet is unrelated to the sun.

Quizplus · Accepted Answer

The cross product of the position vector $\mathbf{r}(t)$ and the velocity vector $\mathbf{v}(t)$ being constant implies that the angular momentum of the planet is conserved. This conservation of angular momentum ensures that the motion of the planet remains in a plane, as the direction of the angular momentum vector (which is perpendicular to the plane of motion) does not change.

The Fact That r(t)×v(t)\mathbf { r } ( t ) \times \mathbf { v } ( t )r(t)×v(t)

The Fact That $\mathbf { r } ( t ) \times \mathbf { v } ( t )$