When a satellite going around the earth in a circular orbit of radius `r` and speed `v` loses some of its energy, then

Options

A. `r` and `v` both increases
B. `r` and `v` both decreases
C. `r` will increases and `v` will decreases
D. `r` will decreases and `v` will increases

Answer

To solve the problem of a satellite losing energy while in a circular orbit around the Earth, we need to analyze the relationship between the radius of the orbit (r), the speed of the satellite (v), and the gravitational force acting on it.

### Step-by-Step Solution:

1. **Understanding Orbital Motion**:
A satellite in a circular orbit experiences a gravitational force that provides the necessary centripetal force to keep it in orbit. The gravitational force \( F_g \) acting on the satellite is given by:
\[ F_g = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the satellite.

2. **Centripetal Force Requirement**:
The centripetal force \( F_c \) required to keep the satellite in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
For the satellite to remain in orbit, these two forces must be equal:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]

3. **Deriving the Orbital Speed**:
By simplifying the equation, we can derive the expression for the orbital speed \( v \):
\[ v^2 = \frac{GM}{r} \]
This shows that the speed of the satellite is dependent on the radius of the orbit.

4. **Effect of Energy Loss**:
When the satellite loses energy, it will lose kinetic energy, which will affect its speed. According to the relationship \( v^2 = \frac{GM}{r} \), if the kinetic energy decreases, the speed \( v \) must decrease.

5. **Change in Orbital Radius**:
As the speed decreases, the gravitational force will no longer provide enough centripetal force to maintain the same orbit. Consequently, the satellite will spiral inward, resulting in a decrease in the radius \( r \) of the orbit.

6. **Conclusion**:
Therefore, when a satellite loses energy, both the radius \( r \) and the speed \( v \) of the satellite decrease. The correct option is:
- **Option 2**: \( r \) and \( v \) both decrease.