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 find the ratio of the kinetic energies of two satellites orbiting the Earth at different heights, we can follow these steps:

### Step 1: Write the expression for kinetic energy of a satellite
The kinetic energy (KE) of a satellite in orbit is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the satellite and \( v \) is its orbital speed.

### Step 2: Relate orbital speed to gravitational force
For a satellite in a circular orbit, the gravitational force provides the necessary centripetal force. This can be expressed as:
\[ \frac{m v^2}{r} = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth to the satellite.

### Step 3: Solve for orbital speed
From the equation above, we can simplify to find the orbital speed \( v \):
\[ v^2 = \frac{GM}{r} \]

### Step 4: Substitute \( v^2 \) into the kinetic energy formula
Now, substituting \( v^2 \) into the kinetic energy formula, we get:
\[ KE = \frac{1}{2} m \left(\frac{GM}{r}\right) = \frac{GMm}{2r} \]

### Step 5: Write the kinetic energy for both satellites
Let the masses of both satellites be \( m \) (since they are the same) and their respective orbital radii be \( r_1 \) and \( r_2 \). Thus, the kinetic energies are:
- For satellite 1:
\[ KE_1 = \frac{GMm}{2r_1} \]
- For satellite 2:
\[ KE_2 = \frac{GMm}{2r_2} \]

### Step 6: Find the ratio of the kinetic energies
Now, we can find the ratio of the kinetic energies of the two satellites:
\[ \frac{KE_1}{KE_2} = \frac{\frac{GMm}{2r_1}}{\frac{GMm}{2r_2}} = \frac{r_2}{r_1} \]

### Final Result
Thus, the ratio of the kinetic energies of the two satellites is:
\[ \frac{KE_1}{KE_2} = \frac{r_2}{r_1} \]