What is escape velocity? Derive an expression for it on the surface of the earth

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What is escape velocity? Derive an expression for it on the surface of the earth

Gravitation - Long Question Answer

Question: What is escape velocity? Derive an expression for it on the surface of the earth.
Answer: The minimum velocity with which a body must be thrown upwards in order that it may just escape the gravitational pull of that planet is called escape velocity.
 
Expression for escape velocity: Let M and R be the mass and radius of the earth. Let ‘m’ be mass of the body thrown vertically upwards with escape velocity ‘v’ from the surface of earth. Let the body be at distance ‘x’ from the center of the earth at an instant then the force of attraction between these two masses is
     F = GMm/x2
 
If we displace this mass through small displacement dx, small work, dW is given by,
     dW  = f * dx
           = GMm/x2 * dx
 
The total amount of work done in taking the object from the surface to infinity is calculated by integrating this equation from limit x = R to x = ꚙ, we get
     Wrꚙ = ʃR GmM/ x2 * dx = GmM ʃR x2 dx = GMm/ R  ...… (i)
 
The kinetic energy by which the body is projected so that it just leaves the gravitational pull of the earth is given by,
     K.E. = ½ mve2 ……... (ii)       Where ve is the escape velocity, of earth
 
Equating (I) and (II), we get
½ mve2 = GMm/ R
ve2 = 2gR2/ R
ve2 = 2gR
ve = 2gR   which is the required expression for escape velocity 

Taking g = 9.8m/s, R = 6400km, we get escape velocity for the earth is 11.2km/sec.

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