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Armeria
Joined: 13 Jun 2009 Posts: 1 Location: New Jersey

Posted: Sun Jun 14, 2009 12:11 am Post subject: speed at perihelion and at aphelion 


An object orbits a star in an elliptical orbit. The distance at the aphelion is 2a and the distance at perihelion is a. Determine the ratio of the objects speed at perihelion to that at aphelion 

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hepcj Site Admin
Joined: 23 Jun 2007 Posts: 125

Posted: Mon Jul 06, 2009 7:44 pm Post subject: 


At aphelion and at perihelion, the velocity is perpendicular to the radial line, that is, theta = 90°. Using the conservation of angular momentum that is implied by Kepler's second Law  equal areas are swept out in equal time.
L = 1/2 r v sin (theta). At aphelion and perihelion theta is 90 deg.
Therefore,
r_p v_p = r_a v_a
(2a) v_p = (a) v_a
where the subscripts p and a refer to perihelion and aphelion, respectively. The ratio the velocity perihelion to the velocity at aphelion is
(v_p) /(v_a) = (r_a)/(r_p)
(v_p)/(v_a) = 2a/a = 2 

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