The radius of a planet is 1/4th of `R_(c )` and its acc accleration due to gravity is 2 g. What would be the value of escape velocity on the planet, if escape velocity on earth is V.

Options

A. `(V)/(sqrt(2))`
B. `Vsqrt(2)`
C. `2 V `
D. `V//2`

Answer

To solve the problem, we need to find the escape velocity on the planet given its radius and acceleration due to gravity. Let's break it down step by step.

### Step 1: Understand the relationship between escape velocity, gravitational acceleration, and radius.
The escape velocity \( V \) from the surface of a planet is given by the formula:
\[ V = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity at the surface of the planet, and \( R \) is the radius of the planet.

### Step 2: Substitute the values for the planet.
According to the problem:
- The radius of the planet \( R_p = \frac{1}{4} R_e \) (where \( R_e \) is the radius of the Earth).
- The acceleration due to gravity on the planet \( g_p = 2g_e \) (where \( g_e \) is the acceleration due to gravity on Earth).

### Step 3: Substitute \( g_p \) and \( R_p \) into the escape velocity formula.
Now we can substitute these values into the escape velocity formula:
\[ V_p = \sqrt{2g_p R_p} \]
Substituting the values we have:
\[ V_p = \sqrt{2(2g_e)\left(\frac{1}{4}R_e\right)} \]

### Step 4: Simplify the expression.
Now simplify the expression:
\[ V_p = \sqrt{2 \cdot 2g_e \cdot \frac{1}{4}R_e} = \sqrt{\frac{4g_e R_e}{4}} = \sqrt{g_e R_e} \]

### Step 5: Relate this to the escape velocity on Earth.
We know that the escape velocity on Earth \( V_e \) is:
\[ V_e = \sqrt{2g_e R_e} \]
Now, we can express \( V_p \) in terms of \( V_e \):
\[ V_p = \sqrt{g_e R_e} = \sqrt{\frac{1}{2} \cdot 2g_e R_e} = \frac{V_e}{\sqrt{2}} \]

### Step 6: Final expression for escape velocity on the planet.
Thus, the escape velocity on the planet \( V_p \) can be expressed as:
\[ V_p = \frac{V}{\sqrt{2}} \]

### Conclusion
The value of escape velocity on the planet is \( \frac{V}{\sqrt{2}} \).