Overview of Kepler's Laws Practice Problems

Two drones are flying in circular orbits around a tall structure at a constant altitude. Drone A has a circular orbit with radius (2r) and takes time (4T) to complete one full revolution around the structure. Drone B has a circular orbit with radius (2r + Δr) and takes time [4T + ΔT] to complete one full revolution, where Δr is much smaller than r and ΔT is much smaller than T. Determine the relationship between the changes in their revolution times and path radii.

While exploring the mysterious Planet Zoraxia, you throw a ball vertically upwards with a velocity of 14 m/s and catch it after 3.2 seconds. Meanwhile, your spaceship is maintaining a stable orbit around the planet at an altitude equal to Planet Zoraxia's radius, taking 245 minutes for a full revolution. Determine (i) the mass and (ii) the radius of the enigmatic Planet Zoraxia.

A planet is located at a distance of 30000 light-years from the center of the Milky Way galaxy. Research has confirmed that the planet has a velocity of 190 km/s when moving about the center of the Milky Way. Determine the number of orbits completed by the planet if it came into existence 6 billion years ago. One light year equals the distance light covers in one year at a speed of 3.0 × 10⁸ m/s.

A planet named Zeta orbits a distant star, called Omicron, in the Andromeda Galaxy. Zeta follows an elliptical path with a period of 8.4 years and an eccentricity of 0.76. At its closest approach during 2015, Zeta was merely 90 AU from Omicron ($1\ AU=1.50\times 10^{11}$ m). Calculate the mass m of the star Omicron, assumed to be a typical massive star. Express m in kilograms and in terms of solar masses $m_{solar}$ . Use: Mass of the Sun = $1.989 \times 10^{30} \mathrm{~kg}$.