Gravitational Potential Energy Practice Problems

A 750 kg rocket on its return journey from Mars to Earth reaches its cruise speed soon after launch. At 10⁵ km from Mars' surface, the speed of the rocket is 36000 km/h. At that position, determine i) the rocket's kinetic energy relative to Mars, and ii) the rocket-Mars potential energy. Mars has a radius of 3390 km and a mass of 6.42 × 10²³ kg.

A group of scientists on Mars are planning to send a package to Earth using a powerful launcher. Assume that the package is launched from a point on Mars that is the closest to Earth. For this problem, the rotation of both planets and the orbiting of Mars should be ignored. What is the minimum speed at which a package must be launched in order to reach Earth? Note that the: gravitational constant $=6.67\times{10}^{-11}{\mathrm{\ }m}^3/{kgs}^2$, mass of the Earth $=5.98\times{10}^{24}\mathrm{\ }kg$, mass of Mars $= 6.39\times{10}^{23}\mathrm{\ }kg$, distance between Earth and Mars $=2.23\times{10}^8\ km$, radius of Mars $=3.39\times{10}^6\ m$, and the distance reached by the package from the center of Mars $=5.50\times{10}^{10}\ m$.

A satellite with a mass equal to that of an average car which is about 1500 kg, orbits Jupiter at a radius equivalent to twice Jupiter's radius which is: r_Jupiter = 69.911 × 10⁶ m. Calculate what the gravitational potential energy is. Note that the mass of Jupiter is: M = 1.898 × 10²⁷ kg.

A space station P with a mass m = 8.0 × 10² kg orbits around Mars at an altitude of 3.0 × 10² km while another space station Q with the same mass orbits around Mars at an altitude of 1.8 × 10⁴ km. Find their gravitational potential energies. Assume that the mass and the radius of Mars are 6.4 × 10²³ kg and 3.4 × 10⁶ m respectively.