Escape Velocity

Escape velocity is the minimum speed at which a body's kinetic energy equals the magnitude of its gravitational binding energy, so it can just reach \infty with zero residual speed. Setting $\frac{1}{2}m{v}_e^2=\frac{GMm}{R}$ and using $GM=g{R}^2$ gives the compact result $v_e=\sqrt{2gR}$. Substituting, $v_e=\sqrt{2\times 9.8\times 6.4\times 10^6}=\sqrt{1.254\times 10^8}\approx 1.12\times 10^4$ m/s, that is about 11.2 km/s. The option $7.9$ km/s is actually the orbital speed near the surface, smaller by a factor of $\sqrt{2}$. The option $22.4$ km/s wrongly doubles the escape speed. The option $8.0$ km/s rounds the orbital speed and is again too small. This reproduces the well-known NCERT value for Earth's escape velocity. A noteworthy feature is that escape velocity is independent of the launch direction in the absence of an atmosphere, because gravity is a conservative force and only the total energy, not the path, determines whether the body reaches \infty. The compact form $v_e=\sqrt{2gR}$ is especially convenient because it expresses escape speed entirely through measurable surface quantities $g$ and $R$, avoiding the need for the Earth's mass or $G$. A plausibility check confirms $v_e=\sqrt{2}$ times the orbital speed, matching the expected relationship between the two characteristic speeds.