CBSE Class 11-science Answered

Escape speed derivation

Asked by athirass31 | 03 Nov, 2019, 11:13: AM

Expert Answer

The principle of conservation of energy helps us to get escape speed. Suppose the object did reach infinity

and that its speed there was v_f. The energy of an object is the sum of potential and kinetic energy.

Let W₁ denotes the gravitational potential energy of the object at infinity.

The total energy E_∞of the projectile at infinity then is

begin mathsize 14px style E subscript infinity space equals space W subscript infinity space plus space m fraction numerator v subscript f superscript 2 over denominator 2 end fraction end style

......................(1)

where W_∞is the potential energy at infinity.

If the object was thrown initially with a speed v_i from a point at a distance (h+R_E) from the center of the earth

(R_E = radius of the earth), its energy initially was

begin mathsize 14px style E subscript h plus R subscript E end subscript space equals space 1 half m space v subscript i superscript 2 space minus space G fraction numerator m space M subscript E over denominator open parentheses h plus R subscript E close parentheses end fraction space plus space W subscript infinity end style

.........................(2)

By the principle of energy conservation Eqs. (1) and (2) must be equal. Hence

begin mathsize 14px style space 1 half m space v subscript i superscript 2 space minus space G fraction numerator m space M subscript E over denominator open parentheses h plus R subscript E close parentheses end fraction space equals 1 half m space v subscript f superscript 2 end style

......................(3)

R.H.S. of eqn.(3) is a positive quantity with a minimum value zero hence so must be the L.H.S.
Thus, an object can reach infinity as long as v_i is such that

begin mathsize 14px style space 1 half m space v subscript i superscript 2 space minus space G fraction numerator m space M subscript E over denominator open parentheses h plus R subscript E close parentheses end fraction space greater or equal than space 0 end style

....................(4)

The minimum value of v_i corresponds to the case when the L.H.S. of Eq. (4) equals zero.

Thus, the minimum speed required for an object to reach infinity (i.e. escape from the earth) corresponds to

begin mathsize 14px style space space v subscript i superscript 2 space equals space fraction numerator 2 G space M subscript E over denominator open parentheses h plus R subscript E close parentheses end fraction space end style

If the object is thrown from the surface of the earth, h=0, and we get

begin mathsize 14px style space space v subscript i superscript 2 space equals space fraction numerator 2 G space M subscript E over denominator R subscript E end fraction space end style

Using the relation g = GM_E/ R_E² , we get

begin mathsize 14px style open parentheses v subscript i close parentheses subscript m i n end subscript equals space square root of 2 space g space R subscript E end root space end style

Using the value of g and R_E, numerically v_i-min≈ 11.2 km/s. This is called the escape speed,

sometimes loosely called the escape +velocity.

Answered by Thiyagarajan K | 03 Nov, 2019, 12:13: PM

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