The diagram below shows three closed orbits around a planet of radius R and mass

Question

The diagram below shows three closed orbits around a planet of radius R and mass M. Orbit 2 is circular while orbits 1 and 3 are ellipitical orbits. All three orbits are tangent to one another at point P in the diagram.

Suppose that the eccentricity of orbits 1 and 3 is e= 0.5 and that point P is a distance 6R from the center of the planet. In terms of the gravitational constant G, M, and R, determine expressions for:

a) The speed of a satellite passing point P along orbit 1

b) The speed of a satellite passing point Q along orbit 1

Explanation / Answer

a] speed at P in orbit 1 = velocity at perigee = sqrt(GM*(1+e)/P) where P = 6R,

Vp = sqrt(GM*(1+0.5)/6R)

Vp= sqrt(GM/4R)

b] Q is apogee in orbit 1, speed at apogee =  sqrt(GM*(1-e)/A) where A=P*(1+e)/(1-e) = 6R*1.5/0.5 = 18R

Va =  sqrt(GM*(1-0.5)/18R)

Va= sqrt(GM/36R)

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