Topic 6.6 Notes – Motion of Orbiting Satellites

1. A Two-Object Gravitational System

A satellite-central object system is just two objects interacting only through gravity.

The gravitational force between them is

$$F_g = \frac{GMm}{r^2}$$

• $M$ = mass of the central object
• $m$ = mass of the satellite
• $r$ = distance between their centers
• $G$ = universal gravitational constant

In AP Physics 1, we assume:

• $M \gg m$
• The central object’s motion is negligible.
• We analyze the satellite’s motion around a nearly fixed center.

Gravity is a central force. It always points toward the center. That one fact leads to:

• Curved motion (orbits)
• Conservation of mechanical energy
• Conservation of angular momentum

Those conservation laws are what actually control the motion.

2. Gravitational Potential Energy and Total Mechanical Energy

Gravitational Potential Energy

For two objects interacting gravitationally,

$$U_g = -\frac{GMm}{r}$$

Important ideas:

• $U_g = 0$ at infinite separation
• For any finite $r$, $U_g$ is negative
• As $r$ decreases, $U_g$ becomes more negative
• Energy must be added to separate the objects to infinity

The graph below shows how $U_g$ depends on distance $r$.

In the diagram, notice the Sun is at one focus of the ellipse, not at the center.

What changes:

• Speed (fastest at periapsis)
• Kinetic energy
• Gravitational potential energy

What stays constant:

• Total mechanical energy
• Angular momentum

As the satellite moves closer:

• $U_g$ becomes more negative
• $K$ increases

Energy shifts between kinetic and potential, but the total stays fixed.

4. Angular Momentum in Orbits

Angular momentum about the central object is

$$L = mvr$$

(at points where velocity is perpendicular to radius, like periapsis and apoapsis)

Gravity produces no torque about the center because the force points directly toward it.

So angular momentum is conserved.

That means:

• If $r$ decreases → $v$ must increase
• If $r$ increases → $v$ must decrease

This is the deeper reason satellites speed up when closer. On free-response questions, stating “angular momentum is conserved because gravity is a central force” earns real points.

5. Escape Velocity

Escape velocity is the speed that makes total mechanical energy equal to zero.

Start with

$$0 = \frac{1}{2}mv_{esc}^2 - \frac{GMm}{r}$$

Solve for $v_{esc}$:

$$v_{esc} = \sqrt{\frac{2GM}{r}}$$

Notice:

• Depends on $M$ and $r$
• Does not depend on satellite mass
• It is $\sqrt{2}$ times the circular orbit speed at the same radius

If gravity is the only force and a satellite reaches escape velocity, it keeps moving away forever, slowing down until its speed approaches zero at infinite distance.

That “speed goes to zero at infinity” idea shows up in conceptual questions more than students expect.