Question

Two satellites of equal mass are revolving around earth in elliptical orbits of different semi-major axis. If their angular momentum about earth's centre are in the ratio $3:4$, then the ratio of their areal velocity is

• A. $\frac{3}{4}$
• B. $\frac{2}{3}$
• C. $\frac{1}{3}$
• D. $\frac{4}{3}$

Answer 1

The correct option is A $\frac{3}{4}$

Given:

$\frac{|\overrightarrow{L_1}|}{|\overrightarrow{L_2}|}=\frac{3}{4}$

As we know that areal velocity $v_a$ is given by

$v_a=\frac{\Delta A}{\Delta t}=\frac{|\overrightarrow{L}|}{2m}........\left(1\right)$

$\overrightarrow{L}$ is the angular momentum of satellite and $m$ is the mass of satellite

Here, $m$ is constant. So

$\Rightarrow \frac{v_{a1}}{v_{a2}}=\frac{|\overrightarrow{L_1}|}{|\overrightarrow{L_2}|}=\frac{3}{4}$

Therefore, option $\left(a\right)$ is correct.

Why this question: To make students understand that the areal velocity is dependent on the angular momentum. Tips: It is important to remember the derivation of Kepler’s 2nd and 3rd laws. Areal velocity $=\frac{\Delta A}{\Delta t}=\frac{|\overrightarrow{L}|}{2m}=v_A$