Question

A satellite is revolving around the earth in a circular orbit with kinetic energy of $1.69 \times 10^{10}$ J. The additional kinetic energy required for just escaping into the outer space is

• A. \( 3.38 \times 10^{10} \, \text{J} \)
• B. \( 1.69 \times 10^{10} \, \text{J} \)
• C. \( 0.89 \times 10^{10} \, \text{J} \)
• D. \( 1.35 \times 10^{10} \, \text{J} \)

Hint:

To escape from a satellite orbit, you need to double the kinetic energy already present in the orbit. Keep in mind that the escape velocity for a satellite is related to the current orbital kinetic energy.

Answer 1

The Correct Option is B

For a satellite in a circular orbit, the total mechanical energy is the sum of kinetic and potential energy. The total energy \(E\) is given by: \[ E = \frac{-GMm}{2r} \] Where: 
- \(G\) is the gravitational constant, 
- \(M\) is the mass of the Earth, 
- \(m\) is the mass of the satellite, 
- \(r\) is the radius of the orbit. 
The escape energy is equal to the kinetic energy required to overcome the gravitational pull and is equal to the negative of the potential energy. So, the total kinetic energy needed to escape from orbit is: \[ K_{\text{escape}} = 2K_{\text{orbital}} \] Where \(K_{\text{orbital}}\) is the orbital kinetic energy of the satellite. Since the satellite already has \( 1.69 \times 10^{10} \, \text{J} \) of kinetic energy, the additional energy required to escape is: \[ K_{\text{escape}} = 2 \times 1.69 \times 10^{10} = 3.38 \times 10^{10} \, \text{J} \] 
Thus, the additional kinetic energy required for the satellite to escape into outer space is \( 1.69 \times 10^{10} \, \text{J} \).