91Ó°ÊÓ

First, let's consider the potential energy of the body at its initial height equal to the Earth's radius (i.e., \(2R\) from the center of the Earth). The gravitational potential energy formula is: \[U = - \frac{G M m}{r}\] Where: - \(U\) is the potential energy, - \(G\) is the gravitational constant, - \(M\) is the mass of the Earth, - \(m\) is the mass of the body, and - \(r\) is the distance from the center of the Earth (which is equal to \(2R\)). The gravitational force acting on the body can be calculated using Newton's law of universal gravitation: \[F = \frac{G M m}{r^{2}}\]

Next, we will apply the conservation of energy principle. The initial potential energy of the body will convert into kinetic energy as it falls towards the Earth. The conservation of energy states that the sum of the potential and kinetic energies will be constant: \[U_{i} + K_{i} = U_{f} + K_{f}\] Where: - \(U_{i}\) and \(U_{f}\) are the initial and final potential energies, respectively, - \(K_{i}\) and \(K_{f}\) are the initial and final kinetic energies, respectively. At the initial point, the body is not moving, so its initial kinetic energy is zero: \(K_{i} = 0\). At the final point, when the body reaches the ground, its final potential energy is: \[U_{f} = - \frac{G M m}{R}\] So now the energy conservation equation becomes: \[- \frac{G M m}{2R} = - \frac{G M m}{R} + K_{f}\]

The kinetic energy of the body can be calculated using the formula: \[K = \frac{1}{2} m v^{2}\] Where \(v\) is the velocity of the body. Now, replace \(K_{f}\) in the energy conservation equation with the kinetic energy formula: \[- \frac{G M m}{2R} = - \frac{G M m}{R} + \frac{1}{2} m v^{2}\] Solve for \(v\): \[v^{2} = 2\left(\frac{G M}{R} - \frac{G M}{2R}\right)\] \[v^{2} = 2\left(\frac{G M}{2R}\right)\] \[v = \sqrt{\frac{2 G M}{R}}\] Now, let's compare the calculated velocity with the given options: (A) \(\frac{G M}{R}\) (B) \(\left[\frac{G M}{R}\right]^{1 / 2}\) (C) \(\left[\frac{2 G M}{R}\right]^{1 / 2}\) (D) \(\left[\frac{2 G M}{R}\right]\) Our calculated velocity matches with option (C). Therefore, the correct answer is: \[v = \left[\frac{2 G M}{R}\right]^{1 / 2}\]