A satellite of a mass `m` orbits the earth at a height `h` above the surface of the earth. How much energy must be expended to rocket the satellite out of earth's gravitational influence? (where `M_(E)` and `R_(E)` be mass and radius of the earth respectively)

Options

A. `(GM_(E)m)/(4(R_(E)+h))`
B. `(GM_(E)m)/(2(R_(E)+h))`
C. `(GM_(E)m)/((R_(E)+h))`
D. `(2GM_(E)m)/((R_(E)+h))`

Explanation

To determine the energy required to rocket a satellite of mass \( m \) out of Earth's gravitational influence from a height \( h \) above the surface of the Earth, we can follow these steps:

### Step 1: Understand the Energy at Height \( h \)

The gravitational potential energy \( E_h \) of the satellite at height \( h \) above the Earth's surface can be expressed using the formula:

\[ E_h = -\frac{G M_E m}{R_E + h} \]

where:
- \( G \) is the universal gravitational constant,
- \( M_E \) is the mass of the Earth,
- \( R_E \) is the radius of the Earth,
- \( m \) is the mass of the satellite.

### Step 2: Energy at Infinity

At an infinite distance from the Earth, the gravitational potential energy is defined to be zero:

\[ E_{\infty} = 0 \]

### Step 3: Energy Required to Escape

To find the energy \( E' \) that must be expended to rocket the satellite out of Earth's gravitational influence, we set up the equation based on the conservation of energy. The total energy at height \( h \) plus the energy supplied \( E' \) must equal the energy at infinity:

\[ E_h + E' = E_{\infty} \]

Substituting the known values:

\[ -\frac{G M_E m}{R_E + h} + E' = 0 \]

### Step 4: Solve for \( E' \)

Rearranging the equation to solve for \( E' \):

\[ E' = \frac{G M_E m}{R_E + h} \]

### Conclusion

Thus, the energy that must be expended to rocket the satellite out of Earth's gravitational influence is:

\[ E' = \frac{G M_E m}{R_E + h} \]