Step 1: Understanding the Concept:
Escape velocity ($v_e$) is the speed needed for an object to break free from a planet's gravitational pull and reach infinity with zero kinetic energy.
It is determined by the gravitational potential energy at the surface.
Step 2: Key Formula or Approach:
The formula for escape velocity is:
\[ v_e = \sqrt{\frac{2GM}{R}} \]
where $G$ is the Gravitational constant, $M$ is the mass of the planet, and $R$ is its radius.
This shows that $v_e \propto \sqrt{M}$ and $v_e \propto \frac{1}{\sqrt{R}}$.
Detailed Explanation:
Let the initial escape velocity be $v_{e1} = 10$ km/s for a planet of mass $M$ and radius $R$.
The new conditions for the second planet are:
New Mass $M' = 4M$.
New Radius $R' = R$.
The new escape velocity $v_{e2}$ is:
\[ v_{e2} = \sqrt{\frac{2GM'}{R'}} \]
Substitute $M'$ and $R'$ into the formula:
\[ v_{e2} = \sqrt{\frac{2G(4M)}{R}} \]
Factor out the constant 4:
\[ v_{e2} = \sqrt{4} \cdot \sqrt{\frac{2GM}{R}} \]
\[ v_{e2} = 2 \cdot v_{e1} \]
Substituting the value of $v_{e1} = 10$ km/s:
\[ v_{e2} = 2 \cdot 10 = 20 \text{ km/s} \]
Step 3: Final Answer:
The new escape velocity is 20 km/s.