The ratio of escape velocity of a planet to the escape velocity of earth will be-:
Given: Mass of the planet is`16` times mass of earth and radius of the planet is `4` times the radius of earth .

Options

A. `1:4`
B. `1:sqrt2`
C. `4:1`
D. `2:1`

Answer

To find the ratio of the escape velocity of a planet to the escape velocity of Earth, we will use the formula for escape velocity:

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

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

### Step 1: Write the escape velocity for the planet and Earth

Let:
- \( M_E \) = Mass of Earth
- \( R_E \) = Radius of Earth
- \( M_P \) = Mass of the planet = \( 16 M_E \)
- \( R_P \) = Radius of the planet = \( 4 R_E \)

The escape velocity for Earth \( V_E \) is given by:

\[ V_E = \sqrt{\frac{2GM_E}{R_E}} \]

The escape velocity for the planet \( V_P \) is given by:

\[ V_P = \sqrt{\frac{2GM_P}{R_P}} \]

### Step 2: Substitute the values for the planet

Substituting \( M_P \) and \( R_P \) into the escape velocity formula for the planet:

\[ V_P = \sqrt{\frac{2G(16M_E)}{4R_E}} \]

### Step 3: Simplify the equation

Now simplify the expression:

\[ V_P = \sqrt{\frac{32GM_E}{4R_E}} \]
\[ V_P = \sqrt{\frac{8GM_E}{R_E}} \]

### Step 4: Find the ratio of escape velocities

Now, we can find the ratio of the escape velocity of the planet to that of Earth:

\[ \frac{V_P}{V_E} = \frac{\sqrt{\frac{8GM_E}{R_E}}}{\sqrt{\frac{2GM_E}{R_E}}} \]

### Step 5: Simplify the ratio

Since \( G \) and \( R_E \) are common in both escape velocity expressions, they cancel out:

\[ \frac{V_P}{V_E} = \sqrt{\frac{8}{2}} \]
\[ \frac{V_P}{V_E} = \sqrt{4} \]
\[ \frac{V_P}{V_E} = 2 \]

### Final Answer

Thus, the ratio of the escape velocity of the planet to the escape velocity of Earth is:

\[ \frac{V_P}{V_E} = 2 : 1 \]