Gravity and velocity of a body in an elliptical orbit

[Suraj M]

consider a body revolving around a star and having a velocity v when closest to the star ( distance r) then the velocity of the body at a point farthest ( distance R) is?
1)by angular momentum conservation ::
r × mv = R× mV
»V = (r/R).v

2) By centrifugal and gravitatinal force
v²/r = GM/r²
v² α 1/r
in this we can get V = (r/R)^½ v

the angular momentum method is the right method ... but what am i missing out on from method no. 2??

 

[A.T.]

v²/r

That's the centripetal acceleration for circular motion, not elliptical motion in general.

 

[Suraj M]

That's the centripetal acceleration for circular motion, not elliptical motion in general.

considering a small sector of the particles motion ..can't we use this
if not then how do we solve this through gravitational force??

 

[A.T.]

considering a small sector of the particles motion ..can't we use this

No.

if not then how do we solve this through gravitational force??

Gravitational potential energy vs. kinetic energy?

 

[Suraj M]

No.

Gravitational potential energy vs. kinetic energy?

how'd you do that? Is the sum of the KE and GPE const.?? that's not necessary is it?

 

[A.T.]

Is the sum of the KE and GPE const.?? that's not necessary is it?

If M >> m then it's approximately true in the approximately inertial rest frame of M. Otherwise you have to consider the KE of M too.

 

[Suraj M]

If M >> m then it's approximately true in the approximately inertial rest frame of M. Otherwise you have to consider the KE of M too.

but in that way i would need the mass of the central body to find V ...and that's not a general case..