Question

A satellite moving around the Earth in a circular orbit has kinetic energy \( E \). Then, the minimum amount of energy to be added so that it escapes from the Earth is:

• A. \( \frac{E}{4} \)
• B. \( E \)
• C. \( \frac{E}{2} \)
• D. \( 2E \)

Hint:

The total mechanical energy of a satellite in a stable circular orbit is negative, with its magnitude equal to the kinetic energy. To escape, the satellite must be given energy equal to its current kinetic energy.

Answer 1

The Correct Option is B

Step 1: Understanding the Energy of a Satellite in Orbit - The total mechanical energy of a satellite in a circular orbit is given by: \[ E_{\text{total}} = KE + PE = -\frac{GMm}{2r}. \] - The kinetic energy of the satellite is given by: \[ KE = \frac{GMm}{2r}. \] - The potential energy of the satellite in orbit is: \[ PE = -\frac{GMm}{r}. \] - The total energy of the system is: \[ E_{\text{total}} = KE + PE = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r}. \] - Since we are given that kinetic energy is \( E \), we write: \[ KE = E = \frac{GMm}{2r}. \] Thus, the total energy of the satellite is: \[ E_{\text{total}} = -E. \]
Step 2: Energy Required for Escape - For the satellite to escape Earth's gravitational field, its total energy must be zero (i.e., it must reach infinity with zero velocity).
- The energy required to remove the satellite from orbit is: \[ E_{\text{required}} = 0 - E_{\text{total}}. \] Substituting \( E_{\text{total}} = -E \), \[ E_{\text{required}} = 0 - (-E) = E. \] Thus, the minimum energy required to make the satellite escape from the Earth is: \[ \boxed{E}. \]

Answer 2

Step 1: Understanding the Energy of a Satellite in Orbit

The total mechanical energy of a satellite in a stable circular orbit is the sum of its kinetic and potential energies. These are defined as:

- Kinetic Energy (KE):
\[ KE = \frac{GMm}{2r} \]
- Potential Energy (PE):
\[ PE = -\frac{GMm}{r} \]
- Total Energy (E_total):
\[ E_{\text{total}} = KE + PE = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r} \]

If the satellite’s kinetic energy is given as \( E \), then:
\[ KE = E = \frac{GMm}{2r} \]
Using this, we can express total energy in terms of \( E \):
\[ E_{\text{total}} = -E \]

Step 2: Energy Required for Escape

To make the satellite escape Earth’s gravity completely, it must reach a point infinitely far away with zero residual velocity, meaning its total energy must become zero:
\[ E_{\text{final}} = 0 \]
Therefore, the minimum energy required to escape is the amount needed to raise its total energy from \( -E \) to \( 0 \):
\[ E_{\text{required}} = E_{\text{final}} - E_{\text{total}} = 0 - (-E) = E \]

Final Answer:

Hence, the minimum energy that must be supplied to the satellite in orbit to escape Earth’s gravity is:
\[ \boxed{E} \]