A satellite is in a circular orbit very close to the surface of a planet. At some point it is given an impulse along its direction of motion, causing its velocity to increase `n` times . It now goes into an elliptical orbit. The maximum possible value of `n` for this to occur is

Options

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

Answer

To solve the problem, we need to determine the maximum value of \( n \) such that a satellite, initially in a circular orbit, can still remain in an elliptical orbit after its velocity is increased \( n \) times.

### Step-by-Step Solution:

1. **Determine the Initial Velocity of the Satellite**:
The initial velocity \( V_i \) of a satellite in a circular orbit very close to the surface of a planet is given by:
\[ V_i = \sqrt{\frac{GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.

2. **Determine the Escape Velocity**:
The escape velocity \( V_e \) from the surface of the planet is given by:
\[ V_e = \sqrt{\frac{2GM}{R}} \]

3. **Condition for Elliptical Orbit**:
For the satellite to remain in an elliptical orbit after the velocity is increased \( n \) times, the new velocity \( V_f \) must be less than or equal to the escape velocity:
\[ V_f = n V_i \leq V_e \]

4. **Substituting the Expressions for Velocities**:
Substituting the expressions for \( V_i \) and \( V_e \):
\[ n \sqrt{\frac{GM}{R}} \leq \sqrt{\frac{2GM}{R}} \]

5. **Simplifying the Inequality**:
We can simplify this inequality by squaring both sides (since both sides are positive):
\[ n^2 \frac{GM}{R} \leq \frac{2GM}{R} \]
Canceling \( \frac{GM}{R} \) from both sides (assuming \( GM/R \neq 0 \)):
\[ n^2 \leq 2 \]

6. **Finding the Maximum Value of \( n \)**:
Taking the square root of both sides gives:
\[ n \leq \sqrt{2} \]
Therefore, the maximum possible value of \( n \) is:
\[ n = \sqrt{2} \]

### Conclusion:
The maximum possible value of \( n \) for the satellite to still remain in an elliptical orbit after the impulse is:
\[ \boxed{\sqrt{2}} \]