Two satellite of masses 40kg, 50kg are revolving around earth in different circular orbits of radii `r_(1),r_(2)` such that their kinetic energies are equal. The ratio of `r_(1),r_(2)` is

Options

A. `sqrt(5):sqrt(4)`
B. `16:25`
C. `sqrt(45):sqrt(4)`
D. `25:16`

Explanation

To solve the problem, we need to find the ratio of the radii \( r_1 \) and \( r_2 \) of the orbits of two satellites that have equal kinetic energies.

### Step-by-Step Solution:

1. **Understand the Kinetic Energy Formula**:
The kinetic energy (KE) of a satellite in orbit is given by the formula:
\[ KE = \frac{G M m}{2 r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.

2. **Set Up the Equation for Kinetic Energies**:
For the two satellites with masses \( m_1 = 40 \, \text{kg} \) and \( m_2 = 50 \, \text{kg} \), and their respective orbital radii \( r_1 \) and \( r_2 \), we can write:
\[ KE_1 = \frac{G M m_1}{2 r_1} \]
\[ KE_2 = \frac{G M m_2}{2 r_2} \]
Since the kinetic energies are equal, we have:
\[ \frac{G M m_1}{2 r_1} = \frac{G M m_2}{2 r_2} \]

3. **Cancel Common Terms**:
We can cancel \( G \) and \( M \) from both sides of the equation:
\[ \frac{m_1}{r_1} = \frac{m_2}{r_2} \]

4. **Substitute the Masses**:
Substitute \( m_1 = 40 \, \text{kg} \) and \( m_2 = 50 \, \text{kg} \):
\[ \frac{40}{r_1} = \frac{50}{r_2} \]

5. **Cross Multiply**:
Cross multiplying gives us:
\[ 40 r_2 = 50 r_1 \]

6. **Rearrange to Find the Ratio**:
Rearranging the equation to find the ratio \( \frac{r_1}{r_2} \):
\[ \frac{r_1}{r_2} = \frac{40}{50} = \frac{4}{5} \]

### Final Answer:
The ratio of the radii \( r_1 \) to \( r_2 \) is:
\[ \frac{r_1}{r_2} = \frac{4}{5} \]