Ncert Exercises & Solutions

What is the direction of the areal velocity of the earth around the sun?

Hint: Use Kepler's second law of planetary motion.

Step 1: Find the areal velocity of the earth.

The areal velocity of the earth around the sun is given by

\frac{d A}{d t} = \frac{L}{2 m}

$\frac{d A}{d t} = \frac{L}{2 m}$

where L is the angular momentum and m is the mass of the earth.

But angular momentum,

L = \vec{r} \times \vec{p} = \vec{r} \times m \vec{v}

$L = \vec{r} \times \vec{p} = \vec{r} \times m \vec{v}$

∴

$∴$ Areal velocity,

(\frac{d A}{d t}) = \frac{1}{2 m} (\vec{r} \times m \vec{v}) = \frac{1}{2} (\vec{r} \times \vec{v})

$(\frac{d A}{d t}) = \frac{1}{2 m} (\vec{r} \times m \vec{v}) = \frac{1}{2} (\vec{r} \times \vec{v})$

Step 2: Find the direction of the areal velocity of the earth.

Therefore, the direction of areal velocity

(\frac{d A}{d t})

$(\frac{d A}{d t})$ is in the direction of (

\vec{r} \times \vec{v}

$\vec{r} \times \vec{v}$ ) i.e. perpendicular to the plane containing r and v and directed as given by the right-hand rule.