The ratio of energy required to raise a satellite to a height `h` above the earth surface to that required to put it into the orbit is

Options

A. `R:h`
B. `h:R`
C. `R:2h`
D. `2h:R`

Answer

To find the ratio of energy required to raise a satellite to a height \( h \) above the Earth's surface to that required to put it into orbit, we will follow these steps:

### Step 1: Calculate the Energy Required to Raise the Satellite (E1)
The gravitational potential energy \( U \) at a distance \( r \) from the center of the Earth is given by:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the satellite.

1. **Initial Energy at Earth's Surface (r)**:
\[ U_i = -\frac{GMm}{r} \]

2. **Final Energy at Height \( h \) (r + h)**:
\[ U_f = -\frac{GMm}{r + h} \]

3. **Energy Required to Raise the Satellite (E1)**:
\[ E_1 = U_f - U_i = \left(-\frac{GMm}{r + h}\right) - \left(-\frac{GMm}{r}\right) \]
\[ E_1 = \frac{GMm}{r} - \frac{GMm}{r + h} \]
\[ E_1 = GMm \left(\frac{1}{r} - \frac{1}{r + h}\right) \]
\[ E_1 = GMm \left(\frac{(r + h) - r}{r(r + h)}\right) = \frac{GMmh}{r(r + h)} \]

### Step 2: Calculate the Energy Required to Put the Satellite into Orbit (E2)
The kinetic energy \( K \) required to put the satellite into orbit is given by:
\[ E_2 = \frac{1}{2} mv^2 \]
where \( v \) is the orbital velocity. The orbital velocity at height \( h \) is given by:
\[ v = \sqrt{\frac{GM}{r + h}} \]
Thus,
\[ E_2 = \frac{1}{2} m \left(\sqrt{\frac{GM}{r + h}}\right)^2 = \frac{1}{2} m \frac{GM}{r + h} = \frac{GMm}{2(r + h)} \]

### Step 3: Find the Ratio of E1 to E2
Now we can find the ratio \( \frac{E_1}{E_2} \):
\[ \frac{E_1}{E_2} = \frac{\frac{GMmh}{r(r + h)}}{\frac{GMm}{2(r + h)}} \]
Cancelling \( GMm \) and \( (r + h) \):
\[ \frac{E_1}{E_2} = \frac{2h}{r} \]

### Final Result
Thus, the ratio of energy required to raise a satellite to a height \( h \) above the Earth's surface to that required to put it into orbit is:
\[ \frac{E_1}{E_2} = \frac{2h}{r} \]