Definition of Escape Velocity: The minimum velocity, by which an object is thrown vertically in an upward direction and that object goes out from the gravitation field of the planet and does not come back, is called the escape velocity. Deduction Escape Velocity Expression: Let us consider the following: The mass of the planet =$M$ The radius of the planet = $R$ The mass of the object = $m$ The gravitational force on an object at position $P$ which is a distance $x$ from the surface of the planet $F=G\frac{M m}{x^{2}} \qquad(1)$ The work done by the force to move the object a very small distance $dx$ from position $A$ to $B$ $W=F.dx$ $dw=G\frac{M m}{x^{2}}dx$ The total work done to move the object from the surface of the planet to infinity $\int^{W}_{0}dw= \int^{\infty}_{R}G\frac{M m}{x^{2}}dx$ $\left[ w \right]^{W}_{0}=G M m \int^{\infty}_{R} \frac{dx}{x^{2}}$ On solving the above equation $W=GM m \int^{\infty}_{R} \frac{dx}{x^{2}}$

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