A satellite is in a circular orbit around a planet . Its period of revolution is T, radius of the orbit is R, orbital velocity V and acceleration .a. , then

Options

A. V=at and `a=V^2/R`
B. `V=(2piR)/T` and V=aT
C. `V=(2piR)/T` and `a=V^2/R`
D. `V=1/2aT^2`

Answer

To solve the problem regarding a satellite in a circular orbit around a planet, we can follow these steps:

### Step 1: Understand the parameters
We have a satellite in a circular orbit with:
- Period of revolution: \( T \)
- Radius of the orbit: \( R \)
- Orbital velocity: \( V \)
- Acceleration: \( a \)

### Step 2: Calculate the orbital velocity \( V \)
The orbital velocity of a satellite in a circular orbit can be calculated using the formula:
\[ V = \frac{\text{Distance traveled in one revolution}}{\text{Time taken for one revolution}} = \frac{2\pi R}{T} \]
This formula arises because the satellite travels a distance equal to the circumference of the orbit, which is \( 2\pi R \), in a time \( T \).

### Step 3: Calculate the centripetal acceleration \( a \)
The centripetal acceleration \( a \) of an object moving in a circular path is given by the formula:
\[ a = \frac{V^2}{R} \]
This means that the acceleration is dependent on the square of the velocity divided by the radius of the orbit.

### Step 4: Substitute the expression for \( V \) into the acceleration formula
We can substitute the expression for \( V \) from Step 2 into the acceleration formula:
\[ a = \frac{(V)^2}{R} = \frac{\left(\frac{2\pi R}{T}\right)^2}{R} \]
This simplifies to:
\[ a = \frac{4\pi^2 R}{T^2} \]

### Conclusion
From our calculations, we have:
1. The orbital velocity \( V = \frac{2\pi R}{T} \)
2. The centripetal acceleration \( a = \frac{V^2}{R} \)

Thus, the correct option that describes the relationship between these quantities is:
- Velocity: \( V = \frac{2\pi R}{T} \)
- Acceleration: \( a = \frac{V^2}{R} \)

### Final Answer
The correct option is the one that states:
- Velocity is \( \frac{2\pi R}{T} \) and acceleration is \( \frac{V^2}{R} \).

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