6.6 Motion of Orbiting Satellites

Motion of Orbiting Satellites AP Physics 1 Summary

Orbiting satellites move under gravity alone, and their motion is locked in by conservation laws. In a circular orbit, energy, speed, and angular momentum stay constant; in an elliptical orbit, total mechanical energy and angular momentum stay constant while kinetic and gravitational potential energy trade back and forth. Escape velocity is the speed where the satellite-central-object system has zero total mechanical energy.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculator

Why This Matters for the AP Physics 1 Exam

This topic ties together gravity, energy conservation, and angular momentum, so it shows up in both multiple-choice and free-response reasoning. You are often asked to compare quantities at different points in an orbit (like periapsis versus apoapsis) and explain why something stays constant or changes. On the free-response section, you cannot just name a conservation law to earn credit. You have to walk through the reasoning that connects the law to your conclusion, which is exactly the skill this topic builds.

Key Takeaways

• In a two-object gravitational system where the satellite's mass is tiny compared to the central object, the central object barely moves, so you can focus only on the satellite.
• Circular orbits keep speed, kinetic energy, gravitational potential energy, angular momentum, and total mechanical energy all constant.
• Elliptical orbits keep total mechanical energy and angular momentum constant, but kinetic and gravitational potential energy each change.
• Gravitational potential energy is $U_g = -G\frac{m_1 m_2}{r}$, defined as zero when the satellite is infinitely far away, and it gets more negative as objects get closer.
• A satellite moves fastest at its closest approach and slowest at its farthest point because of conservation of energy and angular momentum.
• Escape velocity, $v_{esc} = \sqrt{\frac{2GM}{r}}$, is the speed where total mechanical energy of the system equals zero.

Motions in Gravitational Systems

Central Object Motion

In a system with a massive central object and a small orbiting satellite, the satellite's mass is negligible compared to the central object's mass.

• Picture a bowling ball and a marble on a trampoline: the marble orbits while the bowling ball barely moves.
• This approximation lets you analyze the satellite's motion without tracking the central body.
• The Sun-Earth system is a good example, since the Sun's position shifts only slightly because of Earth's gravitational pull.

Satellite Orbit Constraints

Conservation laws tell you what stays constant as a satellite moves through space.

Circular orbits keep these constant:

• Total mechanical energy of the system
• Gravitational potential energy
• Angular momentum of the satellite
• Kinetic energy of the satellite

Elliptical orbits keep these constant:

• Total mechanical energy of the system
• Angular momentum of the satellite

In an elliptical orbit, the satellite's kinetic energy and the system's gravitational potential energy both change throughout the orbit.

Gravitational Potential Energy

Gravitational potential energy is stored energy that depends on position in a gravitational field.

• Rolling a ball up a hill gives it potential energy that can later convert back to motion.
• The gravitational potential energy of a satellite-central object system is defined as zero when the satellite is infinitely far from the central object.
• For orbiting objects, this energy becomes more negative as objects get closer together.
• Setting potential energy to zero at infinite separation is a convenient reference point.
• The formula is $U_g = -\frac{GM_1 m_2}{r}$
  • $G$ = gravitational constant
  • $M_1$ = mass of the central object
  • $m_2$ = mass of the satellite
  • $r$ = distance between the central object and the satellite
• The negative sign means work must be done to separate the objects against gravity.

Circular Orbit Conservation

A circular orbit keeps the satellite at a constant distance from the central object, so its speed, kinetic energy, gravitational potential energy, angular momentum, and total mechanical energy all stay constant.

• The distance from the central object does not change.
• Constant speed means constant kinetic energy.
• Fixed distance means constant gravitational potential energy.
• Total mechanical energy, the sum of kinetic and potential energy, stays constant.
• Gravity provides exactly the centripetal force needed to maintain the circular path.

Elliptical Orbit Conservation

Elliptical orbits show energy conversion in action.

• Total mechanical energy and angular momentum stay constant, but kinetic energy and gravitational potential energy each change.
• The satellite moves fastest at periapsis (closest approach) and slowest at apoapsis (farthest point).
• Kinetic energy rises and falls as speed changes.
• Gravitational potential energy changes as distance changes.
• Speed and position change, but the total mechanical energy stays the same, which is conservation of energy in action.

Escape Velocity

Escape velocity is the threshold between staying in orbit and breaking free from gravity.

• Throw a ball too softly and it falls back; throw it hard enough and it never returns.
• When a satellite reaches escape velocity, the mechanical energy of the satellite-central-object system equals zero.
• If gravity is the only force acting, the satellite moves away from the central object until its speed reaches zero at an infinite distance.
• Conservation of energy gives the escape velocity for a central object of mass $M$: $v_{esc} = \sqrt{\frac{2GM}{r}}$
  • $v_{esc}$ = escape velocity
  • $G$ = gravitational constant
  • $M$ = mass of the central object
  • $r$ = distance between the satellite and the central object at the moment of escape

Escape Velocity Derivation

The escape velocity formula comes straight from energy conservation.

• Consider a satellite of mass $m$ orbiting a central object of mass $M$ at distance $r$.
• The total mechanical energy is the satellite's kinetic energy plus the system's gravitational potential energy: $E = \frac{1}{2}mv^2 - \frac{GMm}{r}$
• To just barely escape, total mechanical energy must equal zero when the satellite reaches an infinite distance: $0 = \frac{1}{2}mv_{esc}^2 - \frac{GMm}{r}$
• Solving for $v_{esc}$ gives: $v_{esc} = \sqrt{\frac{2GM}{r}}$
• Conceptually, escape velocity means the satellite has exactly enough energy to climb the "gravity hill" all the way to infinity.

How to Use This on the AP Physics 1 Exam

Free Response

When you justify a claim, do not stop at naming a conservation law. Connect the law to the specific situation. For example, instead of writing "angular momentum is conserved, so it speeds up," explain that with no external torque the satellite's angular momentum $L = mvr$ stays constant, so as $r$ decreases near periapsis, $v$ must increase. That chain of reasoning is what supports a stronger score.

Problem Solving

• Track which quantities are constant before you start. For circular orbits everything is constant; for elliptical orbits only total mechanical energy and angular momentum are.
• Use $U_g = -G\frac{m_1 m_2}{r}$ with the full distance between centers, not just an altitude above a surface.
• Keep the negative sign on gravitational potential energy. It is built into the formula and matters for energy bookkeeping.
• For escape velocity problems, set total mechanical energy to zero and solve. Notice that escape velocity does not depend on the satellite's mass.

Common Trap

Comparing speeds in an elliptical orbit using only energy can get confusing. Pair conservation of energy with conservation of angular momentum to nail down where the satellite is fastest and slowest.

Practice Problem 1: Energy in an Elliptical Orbit

Answer: In an elliptical orbit, the system's total mechanical energy remains constant and the satellite's angular momentum remains constant. The satellite's kinetic energy changes because its speed changes, and the system's gravitational potential energy changes because the distance between the satellite and planet changes. The satellite moves fastest when closest to the planet and slowest when farthest away.

Practice Problem 2: Gravitational Potential Energy

Think about this conceptually:

1. Gravitational potential energy represents the "energy of position" in a gravitational field.
2. It is negative because energy must be added to separate objects to infinity.
3. The farther apart objects are, the less negative the potential energy becomes.

Mathematically:

• Using the potential energy formula: $U_g = -\frac{GM_1m_2}{r}$
• Substituting values: $U_g = -\frac{6.67 \times 10^{-11} \times 5.97 \times 10^{24} \times 1500}{42 \times 10^6} = -1.42 \times 10^{10} \text{ J}$

Practice Problem 3: Escape Velocity

Think about this conceptually:

1. Escape velocity is the minimum speed needed to leave a gravitational field forever.
2. Mars has less mass than Earth, so its escape velocity should be lower.
3. This is why launching spacecraft from Mars would require less fuel than from Earth.

Mathematically:

• Using the escape velocity formula: $v_{esc} = \sqrt{\frac{2GM}{r}}$
• Substituting values: $v_{esc} = \sqrt{\frac{2 \times 6.67 \times 10^{-11} \times 6.42 \times 10^{23}}{3.39 \times 10^6}} = 5.03 \text{ km/s}$
• Compare to Earth's escape velocity of 11.2 km/s, so Mars requires less than half the escape velocity.

Common Misconceptions

• "Speed is constant in every orbit." Only circular orbits have constant speed. In elliptical orbits, the satellite speeds up near the central object and slows down farther away.
• "Gravitational potential energy is positive." With the infinity-equals-zero reference, $U_g$ is negative for any bound satellite and becomes more negative as the satellite gets closer.
• "Heavier satellites need a higher escape velocity." Escape velocity depends on the central object's mass and the distance, not on the satellite's mass. The satellite mass cancels out.
• "Total mechanical energy changes during an elliptical orbit." Total mechanical energy stays constant; kinetic and potential energy just trade back and forth.
• "Gravitational potential energy uses height above the surface." Use the distance from the center of the central object, not the altitude above its surface.
• "Reaching escape velocity means escaping instantly." The satellite still slows down as it climbs away; its speed only reaches zero at an infinite distance.

Related AP Physics 1 Guides

• 6.3 Angular Momentum and Angular Impulse
• 6.4 Conservation of Angular Momentum
• 6.5 Rolling
• 6.1 Rotational Kinetic Energy
• 6.2 Torque and Work