USAAAO 2022, I кръг — Задача 21
USAAAO 2022 - First Round · 5 февруари 2022 г.
Условие
A satellite is in a circular, equatorial orbit, and can fire its engines to accelerate in any of the following directions:
- In the direction of motion
- Against the direction of motion
- Towards Earth, perpendicular to direction of motion (against radial vector)
- Away from Earth, perpendicular to direction of motion (along radial vector)
- Towards the North Celestial Pole (perpendicular to both direction of motion and radial vector)
Consider a small change of velocity in each of these directions. For how many of these maneuvers will the perigee of the orbit decrease?
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5
Отговори
- C
Решение
Applying the vis-viva equation, a burn in the direction of motion increases instantaneous speed, increasing . As the spacecraft still travels in the same direction, its vertical speed is 0 at this instant; therefore the current point is the perigee of the new orbit, and the perigee stays the same.
Similarly, a burn opposite the direction of motion decreases instantaneous speed, decreasing , meaning that the perigee decreases.
The other four burns are perpendicular to the direction of motion, so increase the satellite’s speed, though only slightly. For the burns towards/away from Earth, vertical speed is nonzero at this point. Therefore, after a short time (or a short time ago), the spacecraft is (or was) at a lower position, which means perigee is lower.
For the last burn, they are perpendicular to the radial vector, so vertical speed remains 0 at this point. As the satellite’s speed has slightly increased, the same logic as a small prograde burn applies, meaning perigee stays the same.
Answer: C
Оригинал в Архива: usaaao_first_exam_2022_problems.pdf · официални решения: usaaao_first_exam_2022_solutions-1.pdf