Explain escape energy and provide its definition. Also, explain escape speed.

(N/A) If we throw a stone upwards, it reaches a certain height and then falls back towards the Earth due to gravity.
If it is thrown with a higher initial speed, it reaches a greater height.
If a stone is thrown with a specific initial speed such that it reaches an infinite distance from the Earth's gravitational field, it will never return. In this state, there is no gravitational attraction acting on it from the Earth.
Consider a body at a distance $r$ from the center of the Earth. If the body is stationary, its total energy $E_i$ is:
$E_i = \text{Kinetic Energy} + \text{Potential Energy} = 0 + \left(-\frac{GM_E m}{r}\right) = -\frac{GM_E m}{R_E + h}$
At an infinite distance, the total energy of the body is considered to be zero. If energy equal to $+\frac{GM_E m}{R_E + h}$ is supplied to the body, its total energy becomes zero, and it escapes the Earth's gravitational field. This required energy is known as escape energy.
To provide this energy as kinetic energy, we must give the body an initial speed $v_e$, known as the escape speed:
$\frac{1}{2} m v_e^2 = \frac{GM_E m}{R_E + h}$
Thus, the escape speed is:
$v_e = \sqrt{\frac{2GM_E}{R_E + h}}$
If the body is on the surface of the Earth, $h = 0$, so:
$v_e = \sqrt{\frac{2GM_E}{R_E}} = \sqrt{2gR_E}$