Three satellites orbit the Earth. The first moves in a circular orbit with a radius R 1 , the second in a circular orbit with a radius R 2 , where R 2 R 1 . The third satellite follows an el

Question

Three satellites orbit the Earth. The first moves in a circular orbit with a radius $R_{1}$ , the second in a circular orbit with a radius $R_{2}$ , where $R_{2} > R_{1}$ . The third satellite follows an elliptical path, touching the smaller circular orbit at its perigee in point P and the larger circular orbit at its apogee in point A.

Let $v_{k 1}$ and $v_{k 2}$ be the speeds in the circular orbits of radii $R_{1}$ and $R_{2}$ , respectively.

Let $v_{p}$ and $v_{a}$ be the speeds at the perigee and apogee of the elliptical orbit, respectively.

Let $T_{1}$ and $T_{2}$ be the periods in the circular orbits of radii $R_{1}$ and $R_{2}$ , respectively; $T_{3}$ is the period in the elliptical orbit.

TestHub · Accepted Answer

For the ellipse, semi-major axis a = 2 R 1 ​ + R 2 ​ ​ . At P the speed exceeds the circular speed at R 1 ​ : v p ​ > v k 1 ​ ​ ; at A it is below the circular speed at R 2 ​ : v a ​ < v k 2 ​ ​ ; and v k 2 ​ ​ < v k 1 ​ ​ since v ∝ r ​ 1 ​ . (A) v a ​ < v k 2 ​ ​ < v k 1 ​ ​ < v p ​ - correct. (B) wrong ordering - incorrect. (C) T 1 ​ = 2 π GM R 1 3 ​ ​ ​ - correct (Kepler). (D) T 3 ​ = 2 π GM a 3 ​ ​ = π 2 GM ( R 1 ​ + R 2 ​ ) 3 ​ ​ - correct. Correct options: A, C, D

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Options:(select one or more)

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