Final answer:
The maximum speed of a planet in its orbit around the Sun, given its minimum speed v0 and the minimum (x1) and maximum (x2) distances, can be found by using the conservation of angular momentum, resulting in the formula v_max = v0 * (x1/x2).
Step-by-step explanation:
The orbital mechanics of a planet around the Sun, in particular, deal with the relationship between the distances from the Sun and the orbital speed of the planet. Kepler's second law, which states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time, can be used to describe this relationship. From this law, we know that the planet travels faster when it is closer to the Sun and slower when it is farther away.
Given the minimum distance x1 and maximum distance x2 of the planet from the Sun, we can use the conservation of angular momentum to find the relationship between the minimum speed, v0, and the maximum speed of the planet. Angular momentum L is conserved and is given by L = mvr, where m is the mass of the planet, v is the velocity, and r is the distance from the Sun. Assuming that the mass of the planet remains constant, and that x1 is the minimum distance with minimum speed v0, and x2 is the maximum distance, we can set up the following relationship due to conservation of angular momentum: m * v0 * x1 = m * v_max * x2. Upon simplifying, the maximum velocity v_max is thus given by v_max = v0 * (x1/x2).