Question

A small satellite is in elliptical orbit around the earth as shown in figure. $L$ denotes the magnitude of its angular momentum and $K$ denotes its kinetic energy. If $1$ and $2$ denote two positions of the satellite, then

• A. $L_2=L_1$ , $k_2=k_1$
• B. $L_2=L_1$ , $k_2>k_1$
• C. $L_2>L_1$ , $k_2<k_1$
• D. $L_2=L_1$ , $k_2<k_1$

Answer 1

Answer: (B)

By Kepler's law, when a satellite is moving around the earth on elliptical path, then its angular momentum remains constant
i.e. $L_1=L_2$
$m_1{v}_1{r}_1=m_2{v}_2{r}_2$
But, $m_1=m_2=m$
$\therefore v_1{r}_1=v_2{r}_2$
$\frac{r_1}{r_2}=\frac{v_2}{v_1}\dots \left(i\right)$
Here, $r_1>r_2$
$\frac{r_1}{r_2}>1$
$\therefore$ From Eq. $\left(i\right)$,
$\frac{v_2}{v_1}>1$
$v_2>v_1$
$v_2^2>v_1^2$ or $\frac{1}{2}m{v}_2^2>\frac{1}{2}m{v}_1^2$
$K_2>K_1$ $(\because K=\frac{1}{2}m{v}^2)$