Know that an object in an elliptical orbit travels faster when closer to the Sun and explain this using the conservation of energy

6.1.2 The Solar System

Key Concept: Speed in Elliptical Orbits

When a planet or a comet moves around the Sun in an elliptical orbit, it travels faster when it is closer to the Sun (at the perihelion) and slower when it is farther away (at the aphelion). This is a direct consequence of the conservation of energy.

Analogy: The Roller‑Coaster 🚀

Imagine a roller‑coaster that starts at a high hill. As it rolls down, it speeds up because the gravitational potential energy is converted into kinetic energy. When it reaches the lowest point, it’s moving the fastest. Then it climbs back up, slowing down as it gains potential energy again. The same idea happens with planets: the Sun’s gravity pulls them in, turning potential energy into kinetic energy.

Conservation of Energy in an Orbit

1. Energy is conserved: the total mechanical energy \(E\) of a planet in orbit is constant.
2. Mathematically, \(E = K + U\) where

   • \(K = \frac{1}{2}mv^2\) is kinetic energy.
   • \(U = -\frac{GMm}{r}\) is gravitational potential energy (negative because gravity is attractive).

3. When the planet is closer to the Sun (\(r\) decreases), the potential energy \(U\) becomes more negative. To keep \(E\) constant, the kinetic energy \(K\) must increase, which means the speed \(v\) increases.
4. Conversely, when the planet moves away from the Sun, \(r\) increases, \(U\) becomes less negative, and \(K\) decreases, so the planet slows down.

Illustration with Numbers

Quick Check Questions

• Why does a planet’s speed increase as it approaches the Sun?
• What happens to the planet’s kinetic energy when it moves from perihelion to aphelion?
• Explain how the conservation of energy principle applies to a comet’s orbit.

Summary

The key takeaway is that in an elliptical orbit, speed is inversely related to distance from the Sun because the Sun’s gravitational pull converts potential energy into kinetic energy as the planet moves closer. This beautiful dance is governed by the simple yet powerful law of conservation of energy.