Question
A planet moving along an elliptical orbit is closest to the sun at a distance $${r_1}$$ and farthest away at a distance of $${r_2}.$$ If $${v_1}$$ and $${v_2}$$ are the linear velocities at these points respectively, then the ratio $$\frac{{{v_1}}}{{{v_2}}}$$ is
A.
$$\frac{{{r_2}}}{{{r_1}}}$$
B.
$${\left( {\frac{{{r_2}}}{{{r_1}}}} \right)^2}$$
C.
$$\frac{{{r_1}}}{{{r_2}}}$$
D.
$${\left( {\frac{{{r_1}}}{{{r_2}}}} \right)^2}$$
Answer :
$$\frac{{{r_2}}}{{{r_1}}}$$
Solution :
Apply conservation of angular momentum.
From the law of conservation of angular momentum, $${L_1} = {L_2}$$
\[{\rm{So,}}\,\,m{r_1}{v_1} = m{r_2}{v_2}\,\,\left[ {\begin{array}{*{20}{c}}
{{\rm{where,}}\,m = {\rm{mass \,the \,of \,planet}}}\\
{r = {\rm{radius\, of \,orbit}}}\\
{v{\rm{ }} = {\rm{ velocity \,of \,the \,planet}}}
\end{array}} \right]\]
$${r_1}{v_1} = {r_2}{v_2} \Rightarrow \frac{{{v_1}}}{{{v_2}}} = \frac{{{r_2}}}{{{r_1}}}$$