For a satellite moving in an orbit around the earth, ratio of kinetic energy to potential energy is

Options

A. 2
B. `-1//2`
C. `(1)/(sqrt(2))`
D. `sqrt(2)`

Answer

To find the ratio of kinetic energy (KE) to potential energy (PE) for a satellite moving in an orbit around the Earth, we can follow these steps:

### Step 1: Write the expressions for kinetic energy and potential energy.
The kinetic energy (KE) of a satellite in orbit is given by the formula:
\[ KE = \frac{G M m}{2r} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( r \) is the radius of the orbit.

The potential energy (PE) of the satellite is given by:
\[ PE = -\frac{G M m}{r} \]

### Step 2: Set up the ratio of kinetic energy to potential energy.
We need to find the ratio \( \frac{KE}{PE} \):
\[ \frac{KE}{PE} = \frac{\frac{G M m}{2r}}{-\frac{G M m}{r}} \]

### Step 3: Simplify the ratio.
Now, simplify the ratio:
\[ \frac{KE}{PE} = \frac{G M m}{2r} \cdot \left(-\frac{r}{G M m}\right) \]
The \( G \), \( M \), and \( m \) terms cancel out:
\[ \frac{KE}{PE} = \frac{1}{2} \cdot (-1) = -\frac{1}{2} \]

### Step 4: Final answer.
Thus, the ratio of kinetic energy to potential energy for a satellite in orbit around the Earth is:
\[ \frac{KE}{PE} = -\frac{1}{2} \]

### Conclusion:
The correct answer is that the ratio of kinetic energy to potential energy for a satellite in orbit is \(-\frac{1}{2}\).

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