Question

A planet of mass `m` revolves in elliptical orbit around the sun of mass `M` so that its maximum and minimum distance from the sun equal to `r_(a)` and `r_(p)` respectively. Find the angular momentum of this planet relative to the sun.

Answer 1

By momentum conservation
`L=mv_(1)r_(1)=mv_(2)r_(2)implies v_(1)r_(1)=v_(2)r_(2) implies v_(2)=(r_(1))/(r_(2))v_(1)`
By conservation
`1/2mv_(1)^(2)-(GM_(s)m)/(r_(1))=1/2mv_(2)^(2)-(GM_(s)m)/(r_(2))`
`v_(1)^(2)-v_(2)^(2)=2GM_(s)(1/(r_(1))-1/(r_(2)))`
`v_(1)^(2)-((r_(1))/(r_(2)))^(2)v_(1)^(2)=2GM_(s)((r_(2)-r_(1))/(r_(1)r_(2)))`
`v_(1)=sqrt((2GM_(s)r_(2))/(r_(1)(r_(1)+r_(2))))`
`L=mv_(1)r_(1)=msqrt((2GM_(s)r_(2))/(r_(1)(r_(1)+r_(2)))) r_(1)=msqrt((2GM_(s)r_(1)r_(2))/((r_(1)+r_(2))))`