D.1.4 Energetics of orbits and escape velocity (HL only)

Equipotential Surfaces

What Are Equipotential Surfaces?

Key Properties of Equipotential Surfaces

1. Perpendicular to Field Lines:
   • Equipotential surfaces are always perpendicular to the gravitational field lines.
2. Zero Work Done:
   • Moving a mass along an equipotential surface requires no work because the potential is constant.

Visualizing Equipotential Surfaces

1. Spherical Mass:
   • Equipotential surfaces around a spherical mass, like a planet, are concentric spheres.
2. Uniform Field:
   • In a uniform gravitational field, equipotential surfaces are parallel planes.

Escape Velocity

What Is Escape Velocity?

Deriving the Formula

To escape, an object’s kinetic energy must equal the gravitational potential energy pulling it back.

1. Kinetic Energy: $$E_k=\frac{1}{2}m{v}^2$$
2. Gravitational Potential Energy: $$E_p=\frac{GMm}{r}$$

• Setting these equal gives:

$$\frac{1}{2}m{v}_{\text{esc}}^2=\frac{GMm}{r}$$

• Solving for $v_{\text{esc}}$:

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

Practical Implications

1. No Directional Requirement:
   • Escape velocity is a scalar quantity; it doesn’t depend on direction.
2. Atmospheric Drag:
   • In reality, atmospheric drag increases the energy needed to escape.

Orbital Speed

What Is Orbital Speed?

Deriving the Formula

For an object in circular orbit, the gravitational force provides thecentripetal force:

1. Gravitational Force: $$F_{\text{gravity}}=\frac{GMm}{r^2}$$
2. Centripetal Force: $$F_{\text{centripetal}}=\frac{m{v}^2}{r}$$

Equating these gives:

$$\frac{GMm}{r^2}=\frac{m{v}^2}{r}$$

Solving for $v$:

$$v_{\text{orbital}}=\sqrt{\frac{GM}{r}}$$

Applications

1. Low Earth Orbit (LEO): Satellites in LEO travel at about 7.8 km/s.
2. Geostationary Orbit: Satellites in higher orbits move more slowly, allowing them to match Earth’s rotation.

Atmospheric Drag

What Is Atmospheric Drag?

Impact on Orbits

1. Decrease in Orbit Height:
   • Drag reduces the object’s kinetic energy, causing it to spiral inward to a lower orbit.
2. Increase in Speed:
   • As the object moves to a lower orbit, its speed increases due to the stronger gravitational pull.

Long-Term Effects

1. Orbital Decay:
   • Without corrective measures, atmospheric drag can cause satellites to re-enter the atmosphere and burn up.
2. Energy Conversion:
   • Drag converts kinetic energy into thermal energy, heating the object.

Reflection

How Do These Concepts Interconnect?

1. Equipotential Surfaces: Help visualize energy changes in a gravitational field.
2. Escape Velocity: Determines whether an object can leave the field entirely.
3. Orbital Speed: Keeps objects in stable orbits.
4. Atmospheric Drag: Demonstrates how external forces can disrupt these conditions.