A satellite is revolving around a planet in a circular orbit. What will happen, if its speed is increased from `upsilon_(0)` to
(a) `sqrt(1.5) upsilon_(0)` (b) `2 upsilon_(0)`

Explanation

To solve the problem, we need to analyze the effects of increasing the speed of a satellite in a circular orbit around a planet. We will consider two cases: when the speed is increased to \( \sqrt{1.5} \upsilon_0 \) and when it is increased to \( 2 \upsilon_0 \).

### Step 1: Understand the initial conditions
The satellite is initially revolving around a planet in a circular orbit with speed \( \upsilon_0 \). The gravitational force provides the necessary centripetal force for the satellite's circular motion.

### Step 2: Establish the relationship between gravitational force and centripetal force
The gravitational force 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 planet, \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.

The centripetal force required for circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( v \) is the speed of the satellite.

Setting these two forces equal gives:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)) and rearranging gives:
\[ v^2 = \frac{GM}{r} \]
Thus, the orbital speed \( \upsilon_0 \) is:
\[ \upsilon_0 = \sqrt{\frac{GM}{r}} \]

### Step 3: Analyze the first case \( \sqrt{1.5} \upsilon_0 \)
Now, if the speed is increased to \( \sqrt{1.5} \upsilon_0 \):
\[ v = \sqrt{1.5} \upsilon_0 \]
The new speed is greater than the original orbital speed.

To find out what happens, we need to determine if this speed is greater than the escape velocity \( v_e \). The escape velocity is given by:
\[ v_e = \sqrt{\frac{2GM}{r}} = \sqrt{2} \upsilon_0 \]

Since \( \sqrt{1.5} \approx 1.22 \), we have:
\[ \sqrt{1.5} \upsilon_0 < \sqrt{2} \upsilon_0 \]
This means that the satellite will not escape the gravitational pull of the planet. Instead, it will move into an elliptical orbit.

### Step 4: Analyze the second case \( 2 \upsilon_0 \)
Now, if the speed is increased to \( 2 \upsilon_0 \):
\[ v = 2 \upsilon_0 \]
We compare this with the escape velocity:
\[ 2 \upsilon_0 > \sqrt{2} \upsilon_0 \]
Since \( 2 \upsilon_0 \) is greater than the escape velocity, the satellite will have enough kinetic energy to escape the gravitational influence of the planet.

### Conclusion
(a) When the speed is increased to \( \sqrt{1.5} \upsilon_0 \), the satellite will not escape and will move into an elliptical orbit.

(b) When the speed is increased to \( 2 \upsilon_0 \), the satellite will escape the gravitational pull of the planet.