Answer
When the rocket is very far away from the earth, the rocket's speed is 9998 m/s
Work Step by Step
We can use conservation of energy to find the speed when the rocket is very far from the earth, where we can assume that the gravitational potential energy is zero. Let $M_e$ be the earth's mass and let $M_r$ be the rocket's mass. Let $R_0$ be the radius of the earth.
$K_f+U_f = K_0+U_0$
$\frac{1}{2}M_r~v_f^2+0 = \frac{1}{2}M_r~v_0^2-\frac{G~M_e~M_r}{R_0}$
$v_f^2 = v_0^2-\frac{2~G~M_e}{R_0}$
$v_f = \sqrt{v_0^2-\frac{2~G~M_e}{R_0}}$
$v_f = \sqrt{(15,000~m/s)^2-\frac{(2)(6.67\times 10^{-11}~m^3/kg~s^2)(5.98\times 10^{24}~kg)}{6.38\times 10^6~m}}$
$v_f = 9998~m/s$
When the rocket is very far away from the earth, the rocket's speed is 9998 m/s.
