Escape Velocity

Escape velocity is derived from energy conservation by equating kinetic energy to the gravitational binding energy of the projectile, giving $v_e=\sqrt{\frac{2GM}{R}}=\sqrt{2gR}$. Crucially, the projectile mass $m$ appears on both sides of $\frac{1}{2}m{v}_e^2=\frac{GMm}{R}$ and cancels completely, so escape velocity is independent of the projectile's own mass. The expression contains only the Earth's mass $M$ and radius $R$, confirming the correct option. The projectile mass option is wrong precisely because of this cancellation. The direction option is wrong because escape depends only on the magnitude of speed, not the launch angle, since gravity is conservative and energy is path-independent. The shape option is wrong as ideal escape ignores air resistance, where only energy matters. This is the standard NCERT conclusion. A sanity check confirms that a feather and a cannonball need the same escape speed in the absence of air, since gravity accelerates all masses equally.