If orbit velocity of planet is given by `v = G^(a)M^(b)R^(c)`, then

Options

A. `a = (1)/(3),b=(1)/(3),c=-(1)/(3)`
B. `a=(1)/(3),b=(1)/(2),c=-(1)/(2)`
C. `a=(1)/(2), b=-(1)/(2),c=(1)/(2)`
D. `a=(1)/(2),b=-(1)/(2),c=-(1)/(2)`

Explanation

To solve the problem of finding the values of \( a \), \( b \), and \( c \) in the expression for orbital velocity \( v = G^a M^b R^c \), we can follow these steps:

### Step 1: Understand the formula for orbital velocity
The orbital velocity \( v \) of a planet is given by the formula:
\[ v = \sqrt{\frac{GM}{R}} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the orbit.

### Step 2: Rewrite the formula in terms of powers
We can express the formula for orbital velocity in terms of powers of \( G \), \( M \), and \( R \):
\[ v = \left(GM\right)^{1/2} R^{-1/2} \]
This can be rewritten as:
\[ v = G^{1/2} M^{1/2} R^{-1/2} \]

### Step 3: Identify the powers of \( G \), \( M \), and \( R \)
From the expression \( v = G^{1/2} M^{1/2} R^{-1/2} \), we can identify the values of \( a \), \( b \), and \( c \):
- The exponent of \( G \) is \( a = \frac{1}{2} \),
- The exponent of \( M \) is \( b = \frac{1}{2} \),
- The exponent of \( R \) is \( c = -\frac{1}{2} \).

### Step 4: Write the final answer
Thus, the values of \( a \), \( b \), and \( c \) are:
\[ a = \frac{1}{2}, \quad b = \frac{1}{2}, \quad c = -\frac{1}{2} \]

### Summary
The final result is:
- \( a = \frac{1}{2} \)
- \( b = \frac{1}{2} \)
- \( c = -\frac{1}{2} \)

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