what is escape velocity and escape energy with deeply define definition and derivation

Escape velocity and Escape energy

Escape velocity
When we project a body upward then after reaching a certain height, the body return back due to gravitational attraction(g) of the earth. If we go increasing the initial velocity of he body, then it returns back after reaching greater and greater height. Ultimately(finally), at a certain velocity of projection the body will go out the gravitation field of the earth and will never return to the earth. This initial velocity of the body is called the 'escape velocity'.

                                                                                                                                                                                              in simple words- The minimum velocity with which a body should be projected from the surface of the earth so that it escapes the earth gravitational field called the escape velocity of the body.

Escape energy
The kinetic energy given to the body to project it with the escape velocity is called 'escape energy'.

Derivation:

Potential energy
The initial gravitational potential energy between two objects at some separations is:

where:

PE_i is the initial gravitation potential energy.   

R_i is the initial separation between the centers of the object in Km.

M is the mass of attracting object(earth) in Kg. 

m is the mass of escaping object in Kg.

NOTE: Since the work required to move an object from the zero reference point of Potential energy to some point in space is negative, the potential energy at that point is also considered negative.

Kinetic energy

The initial kinetic energy of an object projected at some velocity away from the earth or other astronomical body is:

where:

KE_i is initial kinetic energy.

m is the mass of object in Kg.

v_e is the escape velocity.

NOTE: The velocity given to the body is the escape velocity, therefore we considered v_e (escape velocity) in place of v.

Total energy or Total initial energy:

The total initial energy or the total initial energy is the sum of the potential energy and kinetic energy at the release point.

Final energy:

gravitational fields hypothetically extend to infinity. Thus, if the initial velocity is great enough, the object will travel to an infinite separation and thus, escape the gravitational force.

Potential energy at infinity:

PE∞ = gravitational potential energy at infinity.

R∞ = infinite separation between the centers of the objects. 

we know that R∞ = ∞(infinity), thus PE∞ = 0 

Kinetic energy at infinity:    

KE∞ = the Final kinetic energy..

v∞ = final velocity at infinite distance.

At infinity, the velocity of the object is zero (0); 

i.e. v∞ = 0, Thus KE∞ = 0.

Total final energy:

we know that the kinetic energy is moving upward and the potential energy is acting downward, the total energy at the initial position is

  

Now, by the law of conservation of energy

TE_i= TE ∞

KE_i+ PE_i = 0

substitute values: 

Take a square root of each expression to get,

we know that gravitation acceleration (g) is, 

Now by substitute the value of gR_i² in eq. 1

where, 

value of g = 9.81    m/s², or 9.807 m/s²

R_i= R= radius of earth 6378.1 km

To find the escape velocity from the earth surface:

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