The mass of a planet is six times that of the earth. The radius of the planet is twice that of the earth. It's the escape velocity from the earth is `v`, then the escape velocity from the planet is:

Options

A. `sqrt(3)v`
B. `sqrt(2)v`
C. `sqrt(5)v`
D. `sqrt(12)v`

Explanation

To find the escape velocity from the planet, 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 gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

### Step 1: Write the escape velocity for Earth
Let the mass of the Earth be \( M \) and the radius of the Earth be \( R \). The escape velocity from Earth is:

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

### Step 2: Write the mass and radius of the new planet
According to the problem:
- The mass of the planet is \( 6M \) (six times the mass of Earth).
- The radius of the planet is \( 2R \) (twice the radius of Earth).

### Step 3: Substitute the values into the escape velocity formula for the planet
Now, we can substitute these values into the escape velocity formula for the planet:

\[ v_p = \sqrt{\frac{2G(6M)}{2R}} \]

### Step 4: Simplify the expression
Now, simplify the expression:

\[ v_p = \sqrt{\frac{12GM}{2R}} = \sqrt{\frac{6GM}{R}} \]

### Step 5: Relate it to the escape velocity from Earth
We know that the escape velocity from Earth is:

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

Now, we can express \( v_p \) in terms of \( v \):

\[ v_p = \sqrt{3} \cdot \sqrt{\frac{2GM}{R}} = \sqrt{3} \cdot v \]

### Conclusion
Thus, the escape velocity from the planet is:

\[ v_p = \sqrt{3} \cdot v \]

### Final Answer
The escape velocity from the planet is \( \sqrt{3} \cdot v \).

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