If the escape velocity on the earth.s surface is 11.2 `"kms"^(-1)` , its value on a planet having double the mass and double the radius that of the earth is

Options

A. 5.6 `"kms"^(-1)`
B. 11.2 `"kms"^(-1)`
C. 22.4 `"kms"^(-1)`
D. `11.2 sqrt2 "kms"^(-1)`

Answer

To find the escape velocity on a planet that has double the mass and double the radius of Earth, we can use the formula for escape velocity:

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

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: Write the escape velocity for Earth
For Earth, we denote its mass as \( M \) and its radius as \( R \). The escape velocity on Earth is given by:

\[ v_{e, \text{Earth}} = \sqrt{\frac{2GM}{R}} \]

Given that \( v_{e, \text{Earth}} = 11.2 \, \text{km/s} \).

### Step 2: Write the escape velocity for the new planet
For the new planet, the mass is \( 2M \) and the radius is \( 2R \). The escape velocity for this planet can be expressed as:

\[ v_{e, \text{planet}} = \sqrt{\frac{2G(2M)}{2R}} \]

### Step 3: Simplify the expression
Now, simplify the expression for the escape velocity on the new planet:

\[ v_{e, \text{planet}} = \sqrt{\frac{2G(2M)}{2R}} = \sqrt{\frac{2GM}{R}} \]

### Step 4: Relate the escape velocities
Notice that the expression simplifies to:

\[ v_{e, \text{planet}} = \sqrt{\frac{2GM}{R}} = v_{e, \text{Earth}} \]

### Step 5: Conclusion
Thus, the escape velocity on the planet is equal to the escape velocity on Earth:

\[ v_{e, \text{planet}} = 11.2 \, \text{km/s} \]

### Final Answer
The escape velocity on the planet is \( 11.2 \, \text{km/s} \).

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