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Help me understand this and solve it in easiest way possible

Asked by nipunverma59 | 01 Apr, 2019, 07:24: PM

Expert Answer

We have energy Balance : U_R + E₁ = U_{R+h ...........................(1)}

where U_R = Potential energy at surface of earth = - G M m/R ,

E₁ = Energy utilised to take the satellite to a height h above earth surface,

U_R+h= Potential energy of satellite at height h above earth surface = - G M m /(R+h)

G = Gravitational constant, M = mass of earth, m = mass of satellite, R is radius of earth.

hence eqn.(1) is written as :

begin mathsize 12px style negative fraction numerator G space M space m over denominator R end fraction space plus space E subscript 1 space equals space minus fraction numerator G space M space m over denominator R plus h end fraction space space................... left parenthesis 2 right parenthesis end style

hence

begin mathsize 12px style E subscript 1 space equals space G space M space m space open parentheses 1 over R space minus space fraction numerator 1 over denominator R plus h end fraction close parentheses space equals space fraction numerator G space M space m over denominator R plus h end fraction cross times h over R space.................. left parenthesis 3 right parenthesis end style

If satellite is moving with a speed v in a circular path of radius (R+h) from centre of earth, we have

begin mathsize 12px style m space fraction numerator v squared over denominator R plus h end fraction space equals space G fraction numerator M m over denominator open parentheses R plus h close parentheses squared end fraction space space space space space o r space space space v squared space equals space fraction numerator G space M over denominator open parentheses R space plus space h close parentheses end fraction space................ left parenthesis 4 right parenthesis end style

In the above equation, centripetal force is equated to gravitational force.

Hence using eqn.(4), kinetic energy ,

begin mathsize 12px style E subscript 2 space equals space 1 half m v squared space equals space fraction numerator G space M space m over denominator 2 open parentheses R plus h close parentheses end fraction end style

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

if E₁ = E₂ , then from eqn.(3) and eqn.(4), we get h = R/2 = 6400/2 = 3200 km

Answered by Thiyagarajan K | 01 Apr, 2019, 10:37: PM

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