What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R?

Options

A. `(2GmM)/(3R)`
B. `(GmM)/(2R)`
C. `(GmM)/(3R)`
D. `(5GmM)/(6R)`

Answer

To find the minimum energy required to launch a satellite of mass \( m \) from the surface of a planet of mass \( M \) and radius \( R \) into a circular orbit at an altitude of \( 2R \), we can follow these steps:

### Step 1: Determine the radius of the orbit
The altitude of the orbit is \( 2R \), so the total distance from the center of the planet to the satellite is:
\[ r = R + 2R = 3R \]

### Step 2: Calculate the gravitational potential energy (U) at the orbit
The gravitational potential energy of the satellite in orbit is given by the formula:
\[ U = -\frac{GMm}{r} \]
Substituting \( r = 3R \):
\[ U = -\frac{GMm}{3R} \]

### Step 3: Calculate the kinetic energy (K) at the orbit
The gravitational force provides the necessary centripetal force for the satellite. Thus, we have:
\[ \frac{GMm}{(3R)^2} = \frac{mv^2}{3R} \]
From this, we can solve for \( v^2 \):
\[ \frac{GMm}{9R^2} = \frac{mv^2}{3R} \implies v^2 = \frac{GM}{3R} \]
Now, the kinetic energy is:
\[ K = \frac{1}{2} mv^2 = \frac{1}{2} m \left(\frac{GM}{3R}\right) = \frac{GMm}{6R} \]

### Step 4: Calculate the total energy (E) at the orbit
The total mechanical energy \( E \) of the satellite in orbit is the sum of its kinetic and potential energy:
\[ E = K + U = \frac{GMm}{6R} - \frac{GMm}{3R} \]
To combine these, we can write:
\[ E = \frac{GMm}{6R} - \frac{2GMm}{6R} = -\frac{GMm}{6R} \]

### Step 5: Calculate the initial energy (E_initial) at the surface
The initial potential energy when the satellite is on the surface of the planet is:
\[ U_i = -\frac{GMm}{R} \]
The initial kinetic energy is zero since it starts from rest:
\[ K_i = 0 \]
Thus, the total initial energy is:
\[ E_i = U_i + K_i = -\frac{GMm}{R} \]

### Step 6: Calculate the minimum energy required to launch the satellite
The minimum energy required to launch the satellite is the difference between the total energy at the orbit and the total initial energy:
\[ \Delta E = E - E_i = -\frac{GMm}{6R} - \left(-\frac{GMm}{R}\right) \]
\[ \Delta E = -\frac{GMm}{6R} + \frac{6GMm}{6R} = \frac{5GMm}{6R} \]

### Final Answer
Thus, the minimum energy required to launch the satellite is:
\[ \Delta E = \frac{5GMm}{6R} \]

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