Energy of an Orbiting Satellite
In this section, we explore the concepts of kinetic and potential energy for satellites orbiting around the Earth. The relationship between these energies can be expressed mathematically:
- Kinetic Energy (K.E): For a satellite in a circular orbit, the kinetic energy is given by the formula:
$$ K.E = \frac{1}{2}mv^2 = \frac{1}{2} \frac{G M m}{R + h} $$
where:
- $m$ = mass of the satellite,
- $M$ = mass of the Earth,
- $R$ = radius of Earth,
- $h$ = height above Earth's surface.
- Potential Energy (P.E): The gravitational potential energy of the satellite is defined as:
$$ P.E = - \frac{G M m}{R + h} $$
This indicates that the potential energy is negative when considering gravitational potential energy at infinity to be zero.
- Total Energy (T.E): The total mechanical energy of an orbiting satellite is expressed as:
$$ T.E = K.E + P.E = - \frac{1}{2} \frac{G M m}{R + h} $$
This total energy is always negative, indicating that the satellite remains bound to Earth.
- Elliptic Orbits: In elliptical orbits, both kinetic and potential energies vary with position along the orbit, but the total energy remains constant and negative. The significance of this is that if total energy were positive, the satellite would escape Earth's gravitational pull.
This section emphasizes the stability and energy dynamics of satellites in orbit, which are crucial for understanding orbital mechanics.