Gravitational-Potential-Energy

As previously discussed, for an artificial body launched from Earth's surface to become a satellite in orbit, it must reach a minimum speed known as the first cosmic velocity, given by $v_1 = \sqrt{GM/R} = \sqrt{Rg} = 7.9 \times 10^3 \text{ m/s}$.

To launch an artificial body from Earth's surface such that it escapes the Earth's gravitational pull, it must reach a minimum speed known as the second cosmic velocity.

A planet has a radius of $R = 500$ km and the gravitational acceleration at its surface is $g = 1.0$ m/s$^2$.

The average diameter of Mars is $6.79 \times 10^3$ km, and the average diameter of Earth is $1.28 \times 10^4$ km. The mass of Mars is 0.108 times the mass of Earth. Let $g_E = 9.8$ m/s$^2$.

A satellite of mass $m$ moves in a circular orbit of radius $r$ around a central body of mass $M$. The gravitational potential energy of the system is defined to be zero when the satellite is infinitely far away.

The Earth can be treated as a point mass $M = 6.0 \times 10^{24}$ kg with radius $R_e = 6.4 \times 10^6$ m. An object has mass $m = 1$ kg. Use the universal gravitational constant $G \approx 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$.