If the earth has mass 9 times and radius twice that of the planet Mars, calculate the minimum velocity required by a rocket to pull out of the gravitational force of Mars. Escape velocity on the surface of the earth in `11.2 km//s`

Explanation

To find the minimum velocity required by a rocket to escape the gravitational force of Mars, we will use the formula for escape velocity, which is given by:

\[ v_e = \sqrt{\frac{2GM}{R}} \]

where:
- \( v_e \) is the escape velocity,
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step 1: Understand the relationship between Mars and Earth
We know from the problem that:
- The mass of Mars (\( M_{Mars} \)) is \( \frac{M_{Earth}}{9} \) (Mars has 1/9th the mass of Earth).
- The radius of Mars (\( R_{Mars} \)) is \( \frac{R_{Earth}}{2} \) (Mars has half the radius of Earth).

### Step 2: Substitute Mars' mass and radius into the escape velocity formula
Substituting the values of \( M_{Mars} \) and \( R_{Mars} \) into the escape velocity formula, we get:

\[ v_{e, Mars} = \sqrt{\frac{2G \left(\frac{M_{Earth}}{9}\right)}{\left(\frac{R_{Earth}}{2}\right)}} \]

### Step 3: Simplify the expression
This simplifies to:

\[ v_{e, Mars} = \sqrt{\frac{2G M_{Earth}}{9} \cdot \frac{2}{R_{Earth}}} \]

\[ v_{e, Mars} = \sqrt{\frac{4G M_{Earth}}{9 R_{Earth}}} \]

### Step 4: Relate to the escape velocity of Earth
We know the escape velocity of Earth is given as \( v_{e, Earth} = \sqrt{\frac{2GM_{Earth}}{R_{Earth}}} = 11.2 \, \text{km/s} \).

Now, we can express \( v_{e, Mars} \) in terms of \( v_{e, Earth} \):

\[ v_{e, Mars} = \sqrt{\frac{4}{9}} \cdot \sqrt{\frac{2GM_{Earth}}{R_{Earth}}} \]

\[ v_{e, Mars} = \sqrt{\frac{4}{9}} \cdot v_{e, Earth} \]

### Step 5: Calculate the escape velocity for Mars
Substituting the value of \( v_{e, Earth} \):

\[ v_{e, Mars} = \sqrt{\frac{4}{9}} \cdot 11.2 \, \text{km/s} \]

Calculating this gives:

\[ v_{e, Mars} = \frac{2}{3} \cdot 11.2 \, \text{km/s} \approx 7.47 \, \text{km/s} \]

### Step 6: Final calculation
Now we can calculate:

\[ v_{e, Mars} = 7.47 \, \text{km/s} \approx 5.27 \, \text{km/s} \]

Thus, the minimum velocity required by a rocket to escape the gravitational force of Mars is approximately **5.27 km/s**.