Total energy of rocket at the surface of earth
=K.E.+P.E.
=

1

2

mv2+(

−GMm

R

)
At the highest point, v=0,K.E.=0 and P.E.=−

GMm

(R+h)

Total energy =K.E.+P.E.
=0+(

−GMm

R+h

)
=−

GMm

R+h

According to law of conservation of energy

1

2

mv2−

GMm

R

=−

GMm

(R+h)

or

1

2

v2=

GM

R

−

GM

(R+h)

=

gR2

R

−

gR2

(R+h)

=gR(1−

R

R+h

)
=gR(

h

R+h

)
or v2(R+h)=2gRh
or Rv2=2gRh−v2h
=(2gR−v2 )h
or h=

Rv2

2gR−v2

=

(6.4×106)×(5×103)2

2×9.8×(6.4×106×)−(5×103)2

=1.6×106m
Distance of the rocket from the Earth =6.4×106m+1.6×106m=8×106m