Question

A planet is revolving round the sun. Its distance from the sun at Apogee is $r_{A}$ and that at Perigee is $r_{p}$. The mass of planet and sun is $m$ and $M$ respectively, $v _{ A }$ and $v_{ p }$ is the velocity of planet at Apogee and Perigee respectively and $T$ is the time period of revolution of planet round the sun.
(a) $T ^{2}=\frac{\pi^{2}}{2 Gm }\left( r _{ A }+ r _{ p }\right)^{2}$
(b) $T^{2}=\frac{\pi^{2}}{2 G M}\left(r_{A}+r_{p}\right)^{3}$
(c) $v_{ A }\, r _{ A }= v _{ p }\, r _{ p }$
(d) $v_{ A }< v _{ p }, r _{ A }> r _{ p }$

• A. a, b, c
• B. a, b, d
• C. b, c, d
• D. all

Answer 1

Answer: (C)

$T ^{2}=\frac{4 \pi^{2}}{ GM } a ^{3}$
$T ^{2}=\frac{4 \pi^{2}}{ GM }\left(\frac{ r _{ p }+ r _{ A }}{2}\right)^{3}$
$=\frac{\pi^{2}}{2 GM }\left( r _{ p }+ r _{ A }\right)^{3}$
$m v r = $ constant
$ \Rightarrow vr =$ constant
$\therefore v _{ A } r _{ A }= v _{ p } r _{ p }$
$\therefore r _{ A }> r _{ p } $
$\therefore v _{ A }< v _{ p }$