CBSE Class 11-science Answered

derive an expression for escape velocity and orbital velocity

Asked by vedavidyasvy | 30 Sep, 2018, 01:56: PM

Expert Answer

Gravitational force of attraction provides the centripetal force for the earth to revolve around sun.

Hence we have

begin mathsize 12px style G fraction numerator M cross times m over denominator R squared end fraction space equals space 1 half space fraction numerator m cross times v squared over denominator R end fraction space............ left parenthesis 1 right parenthesis end style

where G is gravitational constant, M is mass of Sun, m is mass of earth, R is orbital radius of earth and v is orbital velocity.

from eqn.(1), we get

begin mathsize 12px style v space equals space square root of fraction numerator G cross times M over denominator R end fraction end root end style

Escape velocity :-

Suppose the vertically projected object reach infinity. Let its speed at infinity is v_f .

The energy of an object is sum of potential and kinetic energy.

Let W_∞ denotes the gravitational potential energy of the object at infinity. The total energy of the projectile at infinity is

E(∞) = W_∞ + (1/2)m×v_f²......................(1)

If the object is thrown initially with a speed v_i from earth surface, its initial total energy is

E(R) = (1/2)m×v_i²- [ G×M×m / R ] + W_∞...............(2)

By energy conversion, (1) and (2) are same.

(1/2)m×v_i²- [ G×M×m / R ] = (1/2)m×v_f² ....................(3)

eqns.(1) and (2) are equated to get eqn.(3) by cancelling out W_∞ on both sides

RHS of eqn.(3) is always greater than or equal to zero to make v_i as escape velocity

hence we get

begin mathsize 12px style v subscript i superscript 2 space greater than space fraction numerator 2 G M over denominator R end fraction space space o r space v subscript i superscript 2 space greater than space 2 g R space space w h e r e space g space equals space fraction numerator G M over denominator R squared end fraction space i s space a c c e l e r a t i o n space d u e space t o space g r a v i t y end style

hence we get escape velocity as

begin mathsize 12px style v subscript i space greater or equal than space square root of 2 cross times g cross times R end root end style

Answered by Thiyagarajan K | 04 Oct, 2018, 01:02: AM

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