Gravitational Potential Energy and Escape Velocity

Question–answer flashcards covering definitions, formulas and concepts on gravitational potential energy, its reference level, variation of g, and the derivation and meaning of escape velocity.

What is gravitational potential energy?

The energy stored in an object because of its position or state relative to Earth’s surface or another reference point.

For heights small compared with Earth’s radius (h ≪ R), what expression is used for the gravitational potential energy of a mass m at height h?

U = mgh, where g is treated as constant.

Why can g be assumed constant in the formula U = mgh?

Because the height h is negligible compared with Earth’s radius, so the variation of g with altitude is insignificant.

What reference point is most appropriate for assigning zero gravitational potential energy in orbital problems?

An infinite distance from Earth, where Earth’s gravitational force and g both become zero.

When U is referenced to infinity, why is gravitational potential energy near Earth negative?

Because work must be done (positive energy added) to move the object from Earth to infinity; thus its energy relative to infinity is less than zero.

What is the general (exact) expression for gravitational potential energy of mass m at height h above Earth’s surface?

U = -GMm / (R + h), where M and R are Earth’s mass and radius and the negative sign indicates that the energy is below the zero level at infinity.

How does the acceleration due to gravity g vary with altitude?

g decreases as the distance from Earth’s center increases (g ∝ 1/(R + h)²).

Using Newton’s third equation of motion, what maximum height s does a ball of initial speed u reach when g is constant?

s = u² / (2g).

Define escape velocity.

The minimum initial velocity an object must have at Earth’s surface to overcome Earth’s gravitational attraction and reach infinity without falling back.

Describe the energy condition used to derive escape velocity.

Set total mechanical energy at Earth’s surface equal to zero at infinity: (1/2) m v_esc² − GMm/R = 0.

What formula results for escape velocity from Earth’s surface?

v_esc = √(2GM/R).

What is the kinetic energy of an object when it reaches infinite distance after being launched with escape velocity?

Zero; all initial kinetic energy has been converted into gravitational potential energy.