A satellite is orbiting at a certain height in a circular orbit. If the mass of the planet is reduced to half the initial value, then the satellite would

Options

A. fall on the planet
B. go to the orbit of smaller radius
C. go to the orbit of larger radius
D. escapes from the planet

Answer

To solve the problem, we need to analyze how the reduction of the planet's mass affects the satellite's motion in its orbit.

### Step-by-Step Solution:

1. **Understanding Orbital Velocity**:
The formula for the orbital velocity \( v \) of a satellite in a circular orbit is given by:
\[ v = \sqrt{\frac{GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the distance from the center of the planet to the satellite.

2. **Initial Conditions**:
Let the initial mass of the planet be \( M \) and the radius of the orbit be \( R \). The initial orbital velocity \( v_0 \) is:
\[ v_0 = \sqrt{\frac{GM}{R}} \]

3. **Change in Mass**:
According to the problem, the mass of the planet is reduced to half, so the new mass \( M' \) is:
\[ M' = \frac{M}{2} \]

4. **New Orbital Velocity**:
The new orbital velocity \( v' \) with the reduced mass becomes:
\[ v' = \sqrt{\frac{G \cdot \frac{M}{2}}{R}} = \sqrt{\frac{GM}{2R}} = \frac{v_0}{\sqrt{2}} \]

5. **Escape Velocity**:
The escape velocity \( v_e \) from the planet is given by:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
With the new mass, the escape velocity \( v_e' \) becomes:
\[ v_e' = \sqrt{\frac{2G \cdot \frac{M}{2}}{R}} = \sqrt{\frac{GM}{R}} = v_0 \]

6. **Comparison of Velocities**:
Now we compare the new orbital velocity \( v' \) and the new escape velocity \( v_e' \):
\[ v' = \frac{v_0}{\sqrt{2}} \quad \text{and} \quad v_e' = v_0 \]
Since \( v' < v_e' \), the satellite's orbital velocity is less than the escape velocity.

7. **Conclusion**:
Since the satellite's new orbital velocity is less than the escape velocity, it will no longer be able to maintain its orbit and will eventually fall back to the planet.

### Final Answer:
The satellite would fall back to the planet.