a satelite of mass m is revolving in an orbit of radius R around planet of mass M. it lost some energy against the working air resistance and to an orbit of radius R/2 . the lost energy is

When a satellite in orbit loses energy due to air resistance, it can spiral inward to a lower orbit. In your scenario, we have a satellite of mass \( m \) initially orbiting at a radius \( R \) around a planet of mass \( M \). After losing energy, it moves to an orbit of radius \( R/2 \). To find the lost energy, we can analyze the gravitational potential energy and the kinetic energy of the satellite in both orbits.

Understanding Orbital Mechanics

In orbital mechanics, the total mechanical energy of a satellite in a circular orbit is the sum of its kinetic energy (KE) and gravitational potential energy (PE). The formulas for these energies are as follows:

• Kinetic Energy (KE): \( KE = \frac{1}{2} mv^2 \)
• Gravitational Potential Energy (PE): \( PE = -\frac{GMm}{r} \)

Where \( G \) is the gravitational constant, \( m \) is the mass of the satellite, \( M \) is the mass of the planet, and \( r \) is the radius of the orbit.

Energy in the Initial Orbit

For the initial orbit at radius \( R \), the satellite's speed \( v \) can be derived from the balance of gravitational force and centripetal force:

Using \( \frac{GMm}{R^2} = \frac{mv^2}{R} \), we can simplify to find:

\( v = \sqrt{\frac{GM}{R}} \)

Now, substituting this into the kinetic energy formula:

\( KE_1 = \frac{1}{2} m \left(\sqrt{\frac{GM}{R}}\right)^2 = \frac{GMm}{2R} \)

For the potential energy in the initial orbit:

\( PE_1 = -\frac{GMm}{R} \)

The total energy \( E_1 \) in the initial orbit is:

\( E_1 = KE_1 + PE_1 = \frac{GMm}{2R} - \frac{GMm}{R} = -\frac{GMm}{2R} \)

Energy in the New Orbit

Now, let’s calculate the energies for the new orbit at radius \( R/2 \).

The speed in the new orbit is:

\( v = \sqrt{\frac{GM}{R/2}} = \sqrt{\frac{2GM}{R}} \)

Thus, the kinetic energy in the new orbit is:

\( KE_2 = \frac{1}{2} m \left(\sqrt{\frac{2GM}{R}}\right)^2 = \frac{GMm}{R} \)

For the potential energy in the new orbit:

\( PE_2 = -\frac{GMm}{R/2} = -\frac{2GMm}{R} \)

The total energy \( E_2 \) in the new orbit is:

\( E_2 = KE_2 + PE_2 = \frac{GMm}{R} - \frac{2GMm}{R} = -\frac{GMm}{R} \)

Calculating the Lost Energy

The energy lost by the satellite during its transition from the initial orbit to the new orbit can be calculated by finding the difference between the total energies of the two orbits:

\( \text{Lost Energy} = E_1 - E_2 \)

Substituting the values we found:

\( \text{Lost Energy} = \left(-\frac{GMm}{2R}\right) - \left(-\frac{GMm}{R}\right) \)

\( \text{Lost Energy} = -\frac{GMm}{2R} + \frac{GMm}{R} = \frac{GMm}{2R} \)

Thus, the energy lost by the satellite as it moves to the lower orbit is \( \frac{GMm}{2R} \).

Final Thoughts

This analysis illustrates how energy changes as a satellite moves to a lower orbit due to energy loss from air resistance. Understanding these principles is crucial in fields like astrophysics and aerospace engineering, where the dynamics of satellite motion play a significant role.