Two satellites of masses of `m_(1)` and `m_(2)(m_(1)gtm_(2))` are revolving round the earth in circular orbits of radius `r_(1)` and `r_(2)(r_(1)gtr_(2))` respectively. Which of the following statements is true regarding their speeds `v_(1)` and `v_(2)`?

Options

A. `v_(1)=v_(2)`
B. `v_(1)gtv_(2)`
C. `v_(1)ltv_(2)`
D. `(v_(1))/(r_(1))=(v_(2))/(v_(2))`

Explanation

To solve the problem regarding the speeds of two satellites revolving around the Earth, we can follow these steps:

### Step 1: Understand the forces acting on the satellites
The satellites are in circular orbits around the Earth. The gravitational force provides the necessary centripetal force for the satellites to maintain their circular motion.

### Step 2: Set up the equations for centripetal force and gravitational force
For a satellite of mass \( m \) in a circular orbit of radius \( r \), the centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the speed of the satellite.

The gravitational force \( F_g \) acting on the satellite due to the Earth (mass \( M \)) is given by:
\[ F_g = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant.

### Step 3: Equate the centripetal force to the gravitational force
Since the gravitational force provides the centripetal force, we can set these two forces equal:
\[ \frac{mv^2}{r} = \frac{GMm}{r^2} \]

### Step 4: Simplify the equation
We can cancel \( m \) from both sides (since \( m \neq 0 \)):
\[ \frac{v^2}{r} = \frac{GM}{r^2} \]
Multiplying both sides by \( r \):
\[ v^2 = \frac{GM}{r} \]

### Step 5: Write the speed equations for both satellites
For satellite 1 (mass \( m_1 \), radius \( r_1 \)):
\[ v_1^2 = \frac{GM}{r_1} \]
For satellite 2 (mass \( m_2 \), radius \( r_2 \)):
\[ v_2^2 = \frac{GM}{r_2} \]

### Step 6: Compare the speeds of the two satellites
Now, we can compare \( v_1^2 \) and \( v_2^2 \):
\[ \frac{v_1^2}{v_2^2} = \frac{r_2}{r_1} \]

### Step 7: Analyze the relationship between \( r_1 \) and \( r_2 \)
Given that \( r_1 > r_2 \), it follows that:
\[ \frac{r_2}{r_1} < 1 \]
This implies:
\[ \frac{v_1^2}{v_2^2} < 1 \quad \Rightarrow \quad v_1 < v_2 \]

### Conclusion
Thus, the speed of satellite 1 is less than the speed of satellite 2:
\[ v_1 < v_2 \]

### Final Answer
The correct statement regarding their speeds is that \( v_1 \) is less than \( v_2 \).

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