Question

An object is kept at rest at a distance of 3R above the earth’s surface where $ R $ is earth’s radius. The minimum speed with which it must be projected so that it does not return to earth is: (Assume $ M $ = mass of earth, $ G $ = Universal gravitational constant)

• A. \( \sqrt{\frac{GM}{2R}} \)
• B. \( \sqrt{\frac{GM}{R}} \)
• C. \( \sqrt{\frac{3GM}{R}} \)
• D. \( \sqrt{\frac{2GM}{R}} \)

Hint:

The escape velocity depends on the distance from the center of the Earth. For a distance greater than Earth's radius, adjust the escape velocity formula accordingly.

Answer 1

The Correct Option is A

The minimum speed required for an object to escape the gravitational field of Earth is given by the escape velocity formula: \[ v_{\text{escape}} = \sqrt{\frac{2GM}{R}} \] 
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the distance from the center of the Earth. 
However, in this case, the object is placed at a distance of \( 3R \) from the Earth’s surface. 
The total distance from the center of the Earth is \( 4R \). 
The escape velocity at this distance is: \[ v_{\text{escape}} = \sqrt{\frac{2GM}{4R}} = \sqrt{\frac{GM}{2R}} \] 
Thus, the minimum speed with which the object must be projected is \( \sqrt{\frac{GM}{2R}} \), and the correct answer is (1).