Gravitational Potential Energy and Escape Velocity Worked Examples Cheat Sheet

Key Facts

• Universal gravitational potential energy is $U = -\frac{GMm}{r}$, where $r$ is the center-to-center distance between the two masses.
• Near a planet's surface, the approximation $\Delta U = mg\Delta h$ works only when $\Delta h$ is small compared with the planet's radius.
• The change in gravitational potential energy is $\Delta U = U_f - U_i = -\frac{GMm}{r_f} + \frac{GMm}{r_i}$.
• Total mechanical energy in a gravitational field is $E = K + U = \frac{1}{2}mv^2 - \frac{GMm}{r}$.
• Escape speed from distance $r$ is $v_e = \sqrt{\frac{2GM}{r}}$, and it does not depend on the escaping object's mass $m$.
• For a circular orbit, orbital speed is $v = \sqrt{\frac{GM}{r}}$, which is smaller than escape speed by a factor of $\sqrt{2}$.
• An object escapes if its total mechanical energy is $E \ge 0$, and it remains gravitationally bound if $E < 0$.
• The gravitational field strength at distance $r$ is $g = \frac{GM}{r^2}$, so $g$ decreases as distance from the center increases.

Vocabulary

Gravitational Potential Energy

The energy an object has because of its position in a gravitational field, given by $U = -\frac{GMm}{r}$ for two point masses or spherical bodies.

Escape Velocity

The minimum launch speed needed for an object to reach infinitely far away with no additional propulsion, given by $v_e = \sqrt{\frac{2GM}{r}}$.

Mechanical Energy

The sum of kinetic energy and potential energy, written as $E = K + U$.

Kinetic Energy

The energy of motion of an object, calculated with $K = \frac{1}{2}mv^2$.

Orbital Radius

The distance $r$ from the center of the central body to the orbiting object, not the distance above the surface.

Bound Orbit

A gravitational motion with total mechanical energy $E < 0$, meaning the object does not escape to infinity.