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)`

Explanation

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 expression for Kinetic Energy (KE)
The kinetic energy of a satellite in orbit can be expressed as:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the satellite and \( v \) is its orbital velocity.

### Step 2: Write the expression for Potential Energy (PE)
The gravitational potential energy of the satellite can be expressed as:
\[ PE = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the Earth to the satellite.

### Step 3: Relate the orbital velocity to gravitational force
For a satellite in a stable orbit, the gravitational force provides the necessary centripetal force. Thus, we have:
\[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]
From this, we can solve for \( v^2 \):
\[ v^2 = \frac{G M}{r} \]

### Step 4: Substitute \( v^2 \) into the KE expression
Now, substituting \( v^2 \) back into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\frac{G M}{r}\right) = \frac{G M m}{2r} \]

### Step 5: Find the ratio of KE to PE
Now, we can find the ratio of kinetic energy to potential energy:
\[ \frac{KE}{PE} = \frac{\frac{G M m}{2r}}{-\frac{G M m}{r}} \]

### Step 6: Simplify the ratio
Cancelling out common terms:
\[ \frac{KE}{PE} = \frac{1}{2} \cdot \left(-1\right) = -\frac{1}{2} \]

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

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