Answer
$v_A=6.67\times 10^3m/s$,$v_B=2.77\times 10^3m/s$
Work Step by Step
We can determine the rocket's speed at $A$ and $B$ as follows:
$v_A=\sqrt{\frac{2GMr_a}{r_p(r_p+r_a)}}$
We plug in the known values to obtain:
$v_A=\sqrt{\frac{2\times 66.73(10^{-12})\times 0.6\times 5.97\times 10^{24}}{7.6\times 10^6(7.6\times 10^6+1.38\times 10^6)}}$
This simplifies to:
$v_A=6670m/s=6.67\times 10^3m/s$
Similarly, $v_B=\frac{r_p v_A}{r_a}$
We plug in the known values to obtain:
$v_B=\frac{7.6\times 10^3\times 6670}{13.8\times 10^3}$
This simplifies to:
$v_B=2770m/s=2.77\times 10^3m/s$
