Using energy conservation

K_i + U_i = K_f + U_f

$$ \Rightarrow $$ $${1 \over 2}m{u^2} - {{GmM} \over R}$$ = $${1 \over 2}m{v^2} - {{GmM} \over {2R}}$$

$$ \Rightarrow $$ v = $$\sqrt {{u^2} - {{GM} \over R}} $$

After ejecting a rocket of mass $${m \over {10}}$$ the remaining part of mass $${{9m} \over {10}}$$ will rotate the earth with orbital velocity v₀.

$$ \therefore $$ v₀ = $$\sqrt {{{GM} \over {2R}}} $$

Applying momentum conservation along radial direction,

Before firing rocket momentum of satelite in radial direction = mv

And after firing rocket momentum of satelite in radial direction = 0 and momentum of rocket in radial direction = $${m \over {10}}{v_2}$$

$$ \therefore $$ mv = $${m \over {10}}{v_2}$$

$$ \Rightarrow $$ v₂ = 10v

Now applying momentum conservation along tangential direction we get,

0 = $${m \over {10}}{v_1}$$ - $${{9m} \over {10}}{v_0}$$

$$ \Rightarrow $$$${{9m} \over {10}}{v_0}$$ = $${m \over {10}}{v_1}$$

$$ \Rightarrow $$ v₁ = 9v₀

$$ \therefore $$Total Kinetic Energy of rocket

= $${1 \over 2}{m \over {10}}\left( {v_1^2 + v_2^2} \right)$$

= $${1 \over 2}{m \over {10}}\left( {81v_0^2 + 100{v^2}} \right)$$

= $${m \over {20}}\left( {81\left( {{{GM} \over {2R}}} \right) + 100\left( {{u^2} - {{GM} \over R}} \right)} \right)$$

= $${m \over {20}}\left( {100{u^2} + {{81GM} \over {2R}} - {{100GM} \over R}} \right)$$

= $${m \over {20}}\left( {100{u^2} - {{119GM} \over {2R}}} \right)$$

= $$5m\left( {{u^2} - {{119GM} \over {200R}}} \right)$$