(a) Does it take more energy to get a satellite upto `1500 km` above earth than to put in circular orbit once it is there.
(b) What about `3185 km` ?
(c ) What about `4500 km`? (Take `R_(e) = 6370 km`)

Explanation

To solve the question, we need to analyze the energy required to lift a satellite to a certain height above the Earth and the energy required to put it into orbit at that height. We will do this for three different heights: 1500 km, 3185 km, and 4500 km.

Given:
- Radius of the Earth, \( R_e = 6370 \) km
- Gravitational constant, \( G \)
- Mass of the Earth, \( M \)
- Mass of the satellite, \( m \)

### (a) For a height of 1500 km:

1. **Calculate the change in potential energy (\( \Delta E_1 \)) when raising the satellite to 1500 km:**
\[ \Delta E_1 = -\frac{GMm}{R} + \frac{GMm}{R + H} \]
Here, \( H = 1500 \) km, and \( R = 6370 \) km.
\[ \Delta E_1 = -\frac{GMm}{6370} + \frac{GMm}{6370 + 1500} \]
\[ \Delta E_1 = GMm \left( \frac{1}{6370 + 1500} - \frac{1}{6370} \right) \]

2. **Calculate the change in kinetic energy (\( \Delta E_2 \)) when the satellite is in orbit:**
The orbital velocity \( v \) at height \( H \) is given by:
\[ v = \sqrt{\frac{GM}{R + H}} \]
The kinetic energy is:
\[ \Delta E_2 = \frac{1}{2} mv^2 = \frac{1}{2} m \left( \frac{GM}{R + H} \right) \]
\[ \Delta E_2 = \frac{GMm}{2(R + H)} \]

3. **Compare \( \Delta E_1 \) and \( \Delta E_2 \):**
We need to determine whether \( \Delta E_1 \) is greater than or less than \( \Delta E_2 \).

### (b) For a height of 3185 km:

1. **Calculate \( \Delta E_1 \):**
\[ \Delta E_1 = GMm \left( \frac{1}{6370 + 3185} - \frac{1}{6370} \right) \]

2. **Calculate \( \Delta E_2 \):**
\[ \Delta E_2 = \frac{GMm}{2(6370 + 3185)} \]

3. **Compare \( \Delta E_1 \) and \( \Delta E_2 \):**

### (c) For a height of 4500 km:

1. **Calculate \( \Delta E_1 \):**
\[ \Delta E_1 = GMm \left( \frac{1}{6370 + 4500} - \frac{1}{6370} \right) \]

2. **Calculate \( \Delta E_2 \):**
\[ \Delta E_2 = \frac{GMm}{2(6370 + 4500)} \]

3. **Compare \( \Delta E_1 \) and \( \Delta E_2 \):**

### Final Comparison:
For each height, after calculating \( \Delta E_1 \) and \( \Delta E_2 \), we will determine if it takes more energy to lift the satellite to that height or to put it into orbit.