The radius of the Earth is $R$ and the acceleration due to gravity at the surface of the Earth is $g$ - AQA - A-Level Physics - Question 11 - 2022 - Paper 2

The escape velocity ($v_e$) can be derived from the energy considerations for an object of mass $m$ at the surface of the Earth. The gravitational potential energy ($U$) at a distance $R$ (the radius of the Earth) is given by:

$U = -\frac{GMm}{R}$

The kinetic energy ($K$) needed to escape Earth's gravitational field is equal to the potential energy:

$K = \frac{1}{2}mv_e^2$

Setting these equal gives:

$\frac{1}{2}mv_e^2 = \frac{GMm}{R}$