Question

If \( K \) is the kinetic energy of a satellite at a height \( h \) from the surface of earth, then its total energy is

• A. \( -K \)
• B. \( 2K \)
• C. \( \sqrt{K} \)
• D. \( -\dfrac{K}{2} \)
• E. \( -\dfrac{K}{4} \)

Hint:

For a satellite in circular orbit, always remember: \[ U=-2K \quad \text{and} \quad E=K+U=-K \] These two relations make orbital energy questions very quick to solve.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The total energy (E) of a satellite orbiting the Earth is the sum of its kinetic energy (K) and its gravitational potential energy (U). We need to find the relationship between the total energy and the kinetic energy.
Step 2: Key Formula or Approach:
For a satellite of mass \(m\) orbiting the Earth of mass \(M\) at a distance \(r\) from the center of the Earth, the necessary centripetal force is provided by the gravitational force.
Gravitational Force \(F_g = \frac{GMm}{r^2}\)
Centripetal Force \(F_c = \frac{mv^2}{r}\)
Equating these forces gives the orbital velocity. From there, we can find expressions for kinetic, potential, and total energy.
Kinetic Energy (K) = \(\frac{1}{2}mv^2\)
Potential Energy (U) = \(-\frac{GMm}{r}\)
Total Energy (E) = K + U
Step 3: Detailed Explanation:
First, let's find the expression for kinetic energy. The gravitational force provides the centripetal force for the satellite's orbit:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \] Multiplying both sides by \(r\), we get:
\[ mv^2 = \frac{GMm}{r} \] The kinetic energy \(K\) is given by \(K = \frac{1}{2}mv^2\). Substituting the expression for \(mv^2\):
\[ K = \frac{1}{2} \left( \frac{GMm}{r} \right) = \frac{GMm}{2r} \] The gravitational potential energy \(U\) of the satellite at a distance \(r\) from the center of the Earth is:
\[ U = -\frac{GMm}{r} \] The total energy \(E\) is the sum of kinetic and potential energy:
\[ E = K + U = \frac{GMm}{2r} + \left( -\frac{GMm}{r} \right) \] \[ E = \frac{GMm}{2r} - \frac{2GMm}{2r} = -\frac{GMm}{2r} \] Now we compare the expression for total energy \(E\) with the expression for kinetic energy \(K\).
We have \(K = \frac{GMm}{2r}\) and \(E = -\frac{GMm}{2r}\).
Therefore, we can see that \(E = -K\).
Step 4: Final Answer:
The total energy of the satellite is the negative of its kinetic energy. So, if the kinetic energy is K, the total energy is -K. This corresponds to option (A).