Question

The mass of a planet is six times that of the earth. The radius of the planet is twice that of the earth. If the escape velocity from the earth is '$V_e$', then the escape velocity from the planet is

• A. $\sqrt{3}V_e$
• B. $\sqrt{2}V_e$
• C. $V_e$
• D. $\sqrt{5}V_e$

Hint:

For scaling problems in gravitation, simply substitute the multipliers directly into the core radical structure! Since mass increases by 6 and radius increases by 2, plug them into $\sqrt{M/R}$ to get $\sqrt{6/2} = \sqrt{3}$ instantly.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Determine how a planet's escape velocity changes when its mass and radius are scaled by given factors.

Step 2: Key Formula or Approach:
Escape velocity v_esc = √(2GM/R). For scaling problems, the change factor is proportional to √(Mass multiplier / Radius multiplier).

Step 3: Detailed Explanation:
The mass increases by a factor of 6, and the radius increases by a factor of 2. Substituting these multipliers directly into the radical: New factor = √(6/2) = √3. Therefore, the escape velocity scales by exactly √3 relative to its original value. This direct substitution into the core radical avoids intermediate algebraic expansions.

Step 4: Final Answer:
The escape velocity changes by a factor of √3.