Final answer:
The area swept out by a planet in a given time remains constant due to Kepler's Second Law and the conservation of angular momentum, with the planet speeding up when closer to the Sun and slowing down when farther away to sweep out equal areas.
Step-by-step explanation:
The reason that the area swept out by a planet in a given period of time remains constant, even as the planet speeds up and slows down, is described by Kepler's Second Law. This law states that a line connecting a planet to the Sun covers equal areas during equal intervals of time.
This happens because as a planet moves closer to the Sun in its elliptical orbit, it speeds up due to the stronger gravitational pull, and as it moves away, the gravitational pull weakens and the planet slows down. This speed variation ensures that the areas swept out (A1, A2, A3, etc.) over equal time periods are always the same, maintaining a constant areal velocity.
The constant area swept out by the planet is also a consequence of the conservation of angular momentum. When a planet is closer to the Sun, it has to move faster to sweep out the same area as when it is farther away, where it moves slower. This correlation between distance, speed, and swept area is a natural outcome of the physical laws governing motion in the solar system.