Suppose you read in the newspaper that a new planet has been found. Its average speed in orbit is \(33 \mathrm{km} / \mathrm{s}\). When it is closest to its star it moves at \(31 \mathrm{km} / \mathrm{s}\), and when it is farthest from its star it moves at \(35 \mathrm{km} / \mathrm{s}\). This story is in error because a. the average speed is far too fast. b. Kepler's third law says the planet has to sweep out equal areas in equal times, so the speed of the planet cannot change. c. Kepler's second law says the planet must move fastest when it is closest, not when it is farthest away. d. using these numbers, the square of the orbital period will not be equal to the cube of the semimajor axis.

Kepler's Second Law, also known as the Law of Equal Areas, is a fundamental concept in orbital mechanics. It states that a planet moves in such a way that it sweeps out equal areas in equal times. This law implies a relationship between the speed of a planet and its distance from the star it orbits.
Specifically, a planet moves fastest when it is closest to its star (at perihelion) and slowest when it is farthest from its star (at aphelion). This is due to the conservation of angular momentum. When a planet is near its star, the gravitational pull is stronger, causing it to move more quickly. Conversely, when it is farther away, the gravitational pull is weaker, resulting in a slower speed.
For example, in the given problem, if the planet's speed when it is closest to the star is 31 km/s and when it is farthest is 35 km/s, it contradicts Kepler's Second Law. The planet should actually move faster than 35 km/s when it is closest to the star and slower than 31 km/s when it is farthest.

Planetary motion refers to the movement of planets around a star, primarily governed by gravitational forces. Johannes Kepler formulated three laws to describe planetary motion around the sun.
The first law states that planets move in elliptical orbits with the sun at one focus. This means the distance between a planet and the sun changes throughout its orbit.
The second law, as discussed, deals with the varying speed of a planet along its orbit. Kepler's third law links the orbital period and the semi-major axis of the elliptical orbit. It states that the square of the period of any planet is proportional to the cube of the semi-major axis of its orbit.
Understanding these laws helps us predict the positions and speeds of planets at different times. They are fundamental to the field of orbital mechanics and have numerous applications in space exploration and astronomy.

Orbital mechanics is the study of the motions of artificial and celestial objects under the influence of forces such as gravity. It applies principles from physics to understand and predict the trajectories of bodies in space.
Kepler's laws are cornerstone principles in orbital mechanics. The first law's elliptical orbits dictate how spacecraft and planets travel around stars or planets. Kepler's second law informs us about the changing speeds at different points in an orbit, and the third law provides a relationship between the orbital period and distance from the central body.
Engineers and scientists use orbital mechanics to design satellite orbits, plan space missions, and understand natural celestial phenomena. Studying the orbits helps ensure satellites remain in proper positions, spacecraft reach their intended destinations, and predictions about planetary positions remain accurate.
For students, a good grasp of orbital mechanics requires understanding the fundamental concepts of gravity and motion and applying mathematical formulations like those in Kepler's laws.