Question

The velocity with which a projectile must be fired so that it escapes earth's gravitation does not depend on
A) Mass of the Earth
B) Mass of the Projectile
C) Radius of the Projectile’s orbit
D) Gravitational constant

Answer 1

Escape velocity is the speed when the total of an object's kinetic energy and its potential energy is equivalent to zero; an object that has attained escape velocity is neither on the surface nor in orbit. With escape velocity in a direction tending away from the massive body's ground, the object will go away from the body, slowing permanently and approaching, but never approaching, zero speed.

Complete step-by-step solution:
In science, escape velocity is the tiniest speed required for an accessible, non-propelled object to leave from the gravitational force of a massive body, that is, to finally reach an infinite length from it. Escape velocity grows with the body's mass and drops with the escaping object's range from its centre.
Once escape velocity is reached, no further impulse requires to be applied to resume its escape. In other words, if provided escape velocity, the object will go away from the other body, constantly slowing, and will asymptotically reach zero speed as the object's distance reaches infinity. Speeds larger than escape velocity maintain a positive speed at infinite range.
For a projectile, to escape from the gravitational pull of the earth, Its Kinetic energy should be equal to the potential energy.
Kinetic Energy of a projectile = $\dfrac{1}{2} m v_{e}^{2}$
Potential Energy of a projectile = $G \dfrac{mM}{R}$
$\dfrac{1}{2} m v_{e}^{2} = G \dfrac{mM}{R}$
$\implies v_{e} = \sqrt{\dfrac{2GM}{R}}$
Where, M is the mass of the Earth.
G is the Gravitational constant.
R is the projectile’s orbit radius.
Hence, escape velocity is dependent on the Mass of the Earth, Radius of the earth and the Gravitational constant. Hence, it is not depending on its own mass.

Note: The minimum escape velocity estimates no friction or atmospheric drag, which would raise the required instantaneous speed to escape the gravitational force. There will be no coming acceleration or extraneous deceleration, which would change the needed instantaneous velocity.