Let the speed of the planet at the perihelion P be vp and the sun-planet distance be rp . Relate (rp, vp) to the corresponding quantities at the aphelion (ra, va)

Question

Let the speed of the planet at the perihelion P be vp and the sun-planet distance be rp . Relate (rp, vp) to the corresponding quantities at the aphelion (ra, va) (A) (rp/ra)=(vp/va) (B) (rp2/ra2)=(va/vp) (C) (vp/va)=(ra/rp) (D) (vp2/va2)=(ra/rp)

Tardigrade Admin · Accepted Answer

At the perihelion rp and vp are mutually perpendicular so ra is normal to vap. The angular momentum is conserved, ∴ m va ra=m vp rp ⇒ (vp/va)=(ra/rp)

Q. Let the speed of the planet at the perihelion $P$ be $v_{p}$ and the sun-planet distance be $r_{p} .$ Relate $(r_{p}, v_{p})$ to the corresponding quantities at the aphelion $(r_{a}, v_{a})$

$\frac{r _{p}}{r _{a}} = \frac{v _{p}}{v _{a}}$

$\frac{r _{p}^{2}}{r _{a}^{2}} = \frac{v _{a}}{v _{p}}$

$\frac{v _{p}}{v _{a}} = \frac{r _{a}}{r _{p}}$

$\frac{v _{p}^{2}}{v _{a}^{2}} = \frac{r _{a}}{r _{p}}$

At the perihelion $r_{p}$ and $v_{p}$ are mutually perpendicular so $r_{a}$ is normal to $v_{a}^{p}$ . The angular momentum
is conserved,
$∴ m v_{a} r_{a} = m v_{p} r_{p} \Rightarrow \frac{v _{p}}{v _{a}} = \frac{r _{a}}{r _{p}}$

Q. Let the speed of the planet at the perihelion P be vp​ and the sun-planet distance be rp​. Relate (rp​,vp​) to the corresponding quantities at the aphelion (ra​,va​)

ra​rp​​=va​vp​​

ra2​rp2​​=vp​va​​

va​vp​​=rp​ra​​

va2​vp2​​=rp​ra​​