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speed at perihelion and at aphelion

 
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Armeria



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

PostPosted: Sun Jun 14, 2009 12:11 am    Post subject: speed at perihelion and at aphelion Reply with quote

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
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Joined: 23 Jun 2007
Posts: 125

PostPosted: Mon Jul 06, 2009 7:44 pm    Post subject: Reply with quote

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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