Match the following columns
`{:(,"Column-I",,"Column-II"),("(A)",underset("satellite in circular orbit")("Time period of an earth"),"(p)","Independent of mass of satellite"),("(B)","Orbital velocity of satellite","(q)","Independent of radius of orbit"),("(C)","Mechanical energy of satellite","(r)","Independent of mass of earth"),(,,"(s)","None"):}`

Explanation

To solve the matching question, we need to analyze each item in Column-I and determine its relationship with the items in Column-II based on the properties of satellites in circular orbits.

### Step-by-Step Solution:

1. **Time period of an Earth satellite in circular orbit (A)**:
- The formula for the time period (T) of a satellite in circular orbit is given by:
\[ T = 2\pi \sqrt{\frac{r^3}{GM}} \]
where \( r \) is the radius of the orbit and \( M \) is the mass of the Earth.
- From this formula, we can see:
- **Independent of mass of satellite**: True, as the mass of the satellite does not appear in the formula. (Matches with (p))
- **Independent of radius of orbit**: False, as T is directly proportional to \( r^{3/2} \).
- **Independent of mass of Earth**: False, as T is inversely proportional to \( \sqrt{M} \).

Thus, **A matches with (p)**.

2. **Orbital velocity of satellite (B)**:
- The formula for the orbital velocity (V) of a satellite is given by:
\[ V = \sqrt{\frac{GM}{r}} \]
- Analyzing this formula:
- **Independent of mass of satellite**: True, as the mass of the satellite does not appear in the formula. (Matches with (p))
- **Independent of radius of orbit**: False, as V is inversely proportional to \( \sqrt{r} \).
- **Independent of mass of Earth**: False, as V is directly proportional to \( \sqrt{M} \).

Thus, **B matches with (p)**.

3. **Mechanical energy of satellite (C)**:
- The mechanical energy (E) of a satellite is given by:
\[ E = -\frac{GMm}{2r} \]
where \( m \) is the mass of the satellite.
- Analyzing this:
- **Independent of mass of satellite**: False, as E depends on \( m \).
- **Independent of radius of orbit**: False, as E is inversely proportional to \( r \).
- **Independent of mass of Earth**: False, as E is directly proportional to \( M \).

Thus, **C matches with (s)** (None).

### Final Matching:
- A → (p)
- B → (p)
- C → (s)

### Summary of Matches:
- (A) → (p)
- (B) → (p)
- (C) → (s)