A satellite is revolving around earth in a circular orbit. At some instant the speed of the satellite is increased `sqrt(2)` times its orbital speed keeping its direction unchanged. Then, the new path of the satellite is :

Options

A. circular
B. straight line
C. elliptical
D. parabolic

Explanation

To solve the problem step by step, we will analyze the situation of the satellite before and after the speed change.

### Step 1: Understand the initial conditions
The satellite is initially in a circular orbit around the Earth. The orbital speed \( V_0 \) of a satellite in a circular orbit is given by the formula:
\[ V_0 = \sqrt{\frac{GM}{R}} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the orbit.

**Hint**: Remember that the orbital speed depends on the gravitational force acting on the satellite and the radius of its orbit.

### Step 2: Calculate the new speed
The problem states that the speed of the satellite is increased by \( \sqrt{2} \) times its orbital speed. Therefore, the new speed \( V' \) can be expressed as:
\[ V' = \sqrt{2} \cdot V_0 = \sqrt{2} \cdot \sqrt{\frac{GM}{R}} = \sqrt{\frac{2GM}{R}} \]

**Hint**: When multiplying the speed by \( \sqrt{2} \), ensure to apply it to the entire expression for the orbital speed.

### Step 3: Compare the new speed with escape velocity
The escape velocity \( V_e \) from the surface of the Earth is given by:
\[ V_e = \sqrt{\frac{2GM}{R}} \]
Notice that the new speed \( V' \) is equal to the escape velocity:
\[ V' = \sqrt{\frac{2GM}{R}} = V_e \]

**Hint**: The escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body.

### Step 4: Determine the new path of the satellite
When a satellite reaches escape velocity, it no longer remains in a bound orbit and will follow a parabolic trajectory. This is because the gravitational force is no longer sufficient to keep it in a circular orbit.

**Hint**: Recall that the nature of the trajectory (circular, elliptical, parabolic, or hyperbolic) depends on the speed of the object relative to the escape velocity.

### Conclusion
Since the new speed of the satellite equals the escape velocity, the new path of the satellite will be parabolic.

**Final Answer**: The new path of the satellite is parabolic.