One or more options correct
A satellite of mass moves in a circular orbit of radius about the Earth of radius and surface gravity . A thruster fires briefly and tangentially, reducing the satellite's speed to exactly half its circular-orbit speed ; the direction of motion is unchanged. Treat the Earth as a uniform sphere and ignore the atmosphere. Which of the following statements is/are correct?
- A. The satellite goes into an elliptical orbit that does not intersect the Earth's surface.
- B. The semi-major axis of the new Keplerian orbit is .
- C. If the Earth's entire mass were concentrated at its centre (so that the satellite could not hit the surface), the speed at the point of closest approach of the new orbit would be .
- D. The period of the new Keplerian orbit is times the period of the original circular orbit.
Show answer and solution
Answer: B, C
Take . At the burn point the new speed is with , so the specific energy is . Since , the semi-major axis is (B true). The burn point is the apogee, so the perigee distance is : the orbit would end inside the Earth, so the satellite crashes and (A) is false. Angular momentum conservation gives at that point (C true for the point-mass Earth described in that option). Kepler's third law gives , not , so (D) is false.
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