Question

The escape velocity of a body from a planet of mass \(M\) and radius \(R\) is 14 km/s. The escape velocity of the body from another planet having same mass and diameter \(8R\) (in km/s) is

• A. \(7\)
• B. \(10.5\)
• C. \(14\)
• D. \(28\)

Hint:

Escape velocity is inversely proportional to the square root of radius: \(v_e \propto \dfrac{1}{\sqrt{R}}\). Doubling the radius reduces escape velocity by a factor of \(\sqrt{2}\).

Answer 1

The Correct Option is A

Step 1: Escape velocity is given by the formula: \[ v_e = \sqrt{\dfrac{2GM}{R}} \] Step 2: For the first planet: \[ v_{e1} = \sqrt{\dfrac{2GM}{R}} = 14\ \text{km/s} \] Step 3: For the second planet: - Same mass \(M\) - Diameter \(= 8R \Rightarrow\) Radius \(= 4R\) \[ v_{e2} = \sqrt{\dfrac{2GM}{4R}} = \sqrt{\dfrac{1}{4}} . \sqrt{\dfrac{2GM}{R}} = \dfrac{1}{2} . 14 = 7\ \text{km/s} \] % Final Answer \[ \boxed{7} \]