Question 312: Gravitation

Question

The distance of the two planets A and B from the sun are $r_A$ and $r_B$ respectively. Also $r_B$ is equal to $100 r_A$. If the orbital speed of the planet $A$ is ' $v$ ' then the orbital speed of the planet B is

Accepted Answer

The orbital speed of a planet is determined by the formula for circular orbital velocity:
$$ v = \sqrt{\frac{G M}{r}} $$
where:
$ v $ is the orbital speed,
$ G $ is the gravitational constant,
$ M $ is the mass of the central object (in this case, the Sun),
$ r $ is the distance from the center of the planet to the center of the central object (orbital radius).
For two planets $ A $ and $ B $ with radii $ r_A $ and $ r_B $ (where $ r_B = 100 r_A $), the orbital speed for each planet can be set as follows:
For planet A:
$$ v_A = \sqrt{\frac{G M}{r_A}} $$
For planet B:
$$ v_B = \sqrt{\frac{G M}{r_B}} $$
Since $ r_B = 100 r_A $, substitute into the equation for $ v_B $:
$$ v_B = \sqrt{\frac{G M}{100 r_A}} $$
This can be further simplified to:
$$ v_B = \frac{1}{\sqrt{100}} \sqrt{\frac{G M}{r_A}} $$
$$ v_B = \frac{1}{10} \sqrt{\frac{G M}{r_A}} $$
Thus, since $ v_A = \sqrt{\frac{G M}{r_A}} $, it follows that:
$$ v_B = \frac{1}{10} v_A $$
Therefore, the orbital speed of planet B is $\frac{v}{10}$.
So, the correct answer is
Option A
:
$$ \frac{v}{10} $$

Answer

$\frac{\mathrm{v}}{10}$

Answer

$\frac{v}{2}$

Answer

$ \sqrt{2} v$

Answer

10 v

Gravitation

The distance of the two planets A and B from the sun are r_A and r_B respectively. Also r_B is equal to 100 r_A. If the orbital speed of the planet A is ' v ' then the orbital speed of the planet B is

Choose the correct answer: