Gravitational Potential Energy - Escape Velocity

"The minimum velocity with which a body should be thrown vertically upwards from the surface of the Earth so that it escapes the Earth’s gravitational field, is called the escape velocity (v_e) of the body."

\[v_e=\sqrt{\frac{2GM}{R}}\]

• v_e​ = Escape velocity (minimum speed needed to escape Earth’s gravity)
• G = Universal gravitational constant (6.674 × 10⁻¹¹ Nm²/kg²)
• M = Mass of the Earth (or celestial body)
• R = Radius of the Earth (or distance from the centre of the mass to the object)

When you throw an object up, Earth's gravity pulls it back down. The faster you throw it, the higher it goes before falling back. Escape velocity is the special, minimum initial speed required for an object to completely overcome Earth's gravitational pull. If an object is launched with this velocity, it will reach an infinite distance and never return to the planet. This concept is fundamental to understanding space travel and orbital mechanics.

The concept of escape velocity is based on the conservation of energy. For an object to escape the Earth's gravitational field, it must have enough initial kinetic energy to counteract the negative gravitational potential energy it possesses at the surface.

By applying the law of conservation of energy (Initial Total Energy = Final Total Energy):

1. Move the potential energy term to the right side:
   \[\frac{1}{2} m v_e^2 = \frac{G M m}{R}\]
2. Cancel the mass of the object (m) from both sides:
   \[\frac{1}{2} v_e^2 = \frac{G M}{R}\]
3. Multiply both sides by 2:
   \[v_e^2 = \frac{2 G M}{R}\]
4. Take the square root to solve for v_e:
   \[v_e = \sqrt{\frac{2 G M}{R}}\]