Match the following columns. (for a satellite in circular orbit)
`{:(,"Column-I",,"Column-II"),("(A)","Kinetic energy","(p)",-(GMm)/(2r)),("(B)","Potential energy","(q)",sqrt((GM)/(r))),("(C)","Total energy","(r)",-(GMm)/(r)),("(D)","Orbital speed","(s)",(GMm)/(2r)):}`

Answer

To solve the matching question, we need to identify the correct formulas for each physical quantity related to a satellite in a circular orbit. Let's go through each item step by step.

### Step 1: Identify Kinetic Energy (Column I - A)
The kinetic energy (KE) of a satellite in orbit is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
Where \( v \) is the orbital speed. The orbital speed can be derived from the gravitational force acting as the centripetal force:
\[ F = \frac{GMm}{r^2} = \frac{mv^2}{r} \]
From this, we can derive:
\[ v^2 = \frac{GM}{r} \]
Substituting this back into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m \left(\frac{GM}{r}\right) = \frac{GMm}{2r} \]
Thus, we can match:
- (A) Kinetic energy → (p) \(-\frac{GMm}{2r}\)

### Step 2: Identify Potential Energy (Column I - B)
The gravitational potential energy (PE) of a satellite at a distance \( r \) from the center of the Earth is given by:
\[ PE = -\frac{GMm}{r} \]
Thus, we can match:
- (B) Potential energy → (r) \(-\frac{GMm}{r}\)

### Step 3: Identify Total Energy (Column I - C)
The total mechanical energy (E) of the satellite in orbit is the sum of its kinetic and potential energy:
\[ E = KE + PE \]
Substituting the values we found:
\[ E = \frac{GMm}{2r} - \frac{GMm}{r} = -\frac{GMm}{2r} \]
Thus, we can match:
- (C) Total energy → (s) \(-\frac{GMm}{2r}\)

### Step 4: Identify Orbital Speed (Column I - D)
The orbital speed \( v \) we derived earlier is:
\[ v = \sqrt{\frac{GM}{r}} \]
Thus, we can match:
- (D) Orbital speed → (q) \(\sqrt{\frac{GM}{r}}\)

### Final Matches
Now we can summarize the matches:
- (A) Kinetic energy → (p) \(-\frac{GMm}{2r}\)
- (B) Potential energy → (r) \(-\frac{GMm}{r}\)
- (C) Total energy → (s) \(-\frac{GMm}{2r}\)
- (D) Orbital speed → (q) \(\sqrt{\frac{GM}{r}}\)