Two satellites of same mass are orbiting round the earth at heights of `r_1` and `r_2` from the centre of earth. Their kinetic energies are in the ratio of :

Options

A. `r_2/r_1`
B. `r_1/r_2`
C. `r_1/((r_1+r_2))`
D. `r_2/((r_1+r_2))`

Explanation

To solve the problem of finding the ratio of the kinetic energies of two satellites orbiting the Earth at different heights, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces**:
Each satellite experiences a gravitational force that acts as the centripetal force necessary for circular motion. The gravitational force \( F_g \) acting on a satellite of mass \( m \) at a distance \( r \) from the center of the Earth (where the mass of the Earth is \( M \)) is given by:
\[ F_g = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant.

2. **Centripetal Force**:
The centripetal force required to keep the satellite in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the orbital speed of the satellite.

3. **Equating Forces**:
Since the gravitational force provides the necessary centripetal force, we can set these two forces equal:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]

4. **Canceling Mass**:
Since the mass \( m \) of the satellite is the same for both satellites, we can cancel it from both sides:
\[ \frac{GM}{r^2} = \frac{v^2}{r} \]

5. **Rearranging for Velocity**:
Rearranging the equation gives us:
\[ v^2 = \frac{GM}{r} \]

6. **Kinetic Energy**:
The kinetic energy \( K \) of a satellite is given by:
\[ K = \frac{1}{2} mv^2 \]
Substituting \( v^2 \) from the previous step, we get:
\[ K = \frac{1}{2} m \left(\frac{GM}{r}\right) = \frac{GMm}{2r} \]

7. **Finding the Ratio of Kinetic Energies**:
Let \( K_1 \) be the kinetic energy of the first satellite at distance \( r_1 \) and \( K_2 \) be the kinetic energy of the second satellite at distance \( r_2 \):
\[ K_1 = \frac{GMm}{2r_1} \quad \text{and} \quad K_2 = \frac{GMm}{2r_2} \]
To find the ratio of the kinetic energies:
\[ \frac{K_1}{K_2} = \frac{\frac{GMm}{2r_1}}{\frac{GMm}{2r_2}} = \frac{r_2}{r_1} \]

8. **Conclusion**:
The ratio of the kinetic energies of the two satellites is:
\[ \frac{K_1}{K_2} = \frac{r_2}{r_1} \]

### Final Answer:
The kinetic energies of the two satellites are in the ratio \( \frac{K_1}{K_2} = \frac{r_2}{r_1} \).

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