Question

A body which is initially at rest at a height \( R \) above the surface of the Earth of radius \( R \), falls freely towards the Earth. The velocity on reaching the surface of the Earth is:

• A. \( \sqrt{2gR} \)
• B. \( \sqrt{gR} \)
• C. \( \sqrt{\frac{3}{2} gR} \)
• D. \( \sqrt{4gR} \)

Hint:

For free fall from height \( h \), the velocity on reaching the surface can be found using energy conservation: \[ v = \sqrt{2 g h} \] or using gravitational potential.

Answer 1

The Correct Option is B

Step 1: {Apply conservation of energy}
The total energy remains constant: \[ {Increase in kinetic energy} = {Decrease in potential energy} \] Step 2: {Using gravitational potential energy}
\[ \frac{1}{2} mv^2 = mgR \left( \frac{1}{1 + \frac{h}{R}} \right) \] For \( h = R \): \[ mv^2 = mgR \] \[ v = \sqrt{gR} \] Thus, the correct answer is \( \sqrt{gR} \).