Lesson 5.3 • Competency 5.2Complete Lesson

Orbital Mechanics and Satellites

Circular satellite motion, centripetal balance, orbital speed v = √(GM/r), period T = 2π√(r³/GM), Kepler’s Third Law, kinetic energy, potential energy, total mechanical energy, and binding energy.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

Derive expressions for orbital speed v = √(GM/r) and orbital period T = 2π√(r³/GM) for satellites in circular orbits.
Verify Kepler’s Third Law (T² ∝ r³) and use it to solve orbital ratios and calculate central planetary masses.
Calculate kinetic energy (K = GMm/2r), potential energy (U = -GMm/r), total mechanical energy (E = -GMm/2r = -K), and binding energy.

Core Theory & Concepts

When a projectile is launched horizontally at high speed above the atmosphere, its curved trajectory matches the curvature of the Earth, placing it into continuous free fall around the planet—an orbit. The gravitational pull continuously redirects its velocity vector without altering its speed in a circular orbit.

Orbital Speed

Equating gravitational attraction to centripetal force: GMm/r² = m v²/r => v = √(GM/r). Speed is independent of satellite mass.

v = \sqrt{\frac{GM}{r}}

Orbital Period & Kepler’s Third Law

Time for one complete revolution: T = 2πr / v = 2π√(r³ / GM). Thus T² ∝ r³.

T^2 = \left(\frac{4\pi^2}{GM}\right) r^3

Satellite Energy Relations

Kinetic energy K = GMm/(2r). Potential energy U = -GMm/r. Total mechanical energy E = K + U = -GMm/(2r) = -K = U/2. Binding energy B.E. = -E = GMm/(2r).

E = -K = \frac{1}{2}U = -\frac{GMm}{2r}

Common Misconceptions

Misconception

Astronauts float in the ISS because there is zero gravity in space.

Physical Clarification:At the ISS altitude (~400 km), gravitational acceleration is still ~8.8 m s⁻² (about 90% of surface gravity). Astronauts float because both they and the spacecraft are in continuous free fall toward Earth at the exact same acceleration (apparent weightlessness).

Syllabus Topic Outline

Competency 5.2

1Mechanics of circular orbits: gravitational force as centripetal force
2Orbital velocity derivation: v = √(GM/r)
3Orbital period and Kepler’s Third Law: T² = (4π² / GM) r³
4Kinetic, potential, and total energy relationships: E = -K = U/2
5Satellite binding energy and orbital decay due to atmospheric friction
6Apparent weightlessness in orbit and free fall dynamics