The earth is revolving round the sun in an elliptical orbit. If $\frac{O A}{OB} = x$ , the ratio of speeds of earth at $B$ and $A$ will be

Question

The earth is revolving round the sun in an elliptical orbit. If OB O A ​ = x , the ratio of speeds of earth at B and A will be

TestHub · Accepted Answer

According to Kepler's second law (law of areas), the areal velocity of a planet is constant. This implies that the angular momentum of the Earth around the Sun is conserved.

Let $v_{A}$ and $v_{B}$ be the speeds of the Earth at points A and B, respectively.

Let $r_{A} = O A$ and $r_{B} = OB$ be the distances from the Sun (O) to the Earth at points A and B.

By conservation of angular momentum:

$m v_{A} r_{A} = m v_{B} r_{B}$

$v_{A} r_{A} = v_{B} r_{B}$

The ratio of speeds of Earth at B and A is $\frac{v _{B}}{v _{A}}$ .

From the conservation of angular momentum equation:

$\frac{v _{B}}{v _{A}} = \frac{r _{A}}{r _{B}}$

Given that $\frac{O A}{OB} = x$ , which means $\frac{r _{A}}{r _{B}} = x$ .

Therefore, $\frac{v _{B}}{v _{A}} = x$ .

The final answer is $x$ .

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