Answer
$v_{Ac}=5.27\times 10^3m/s$, $\Delta v=684m/s$
Work Step by Step
We can determine the required speed at $A$ and the change in the speed as follows:
$v_{Ac}=\sqrt{\frac{GM_e}{h_{\circ}+r_e}}$
We plug in the known values to obtain:
$v_{Ac}=\sqrt{\frac{66.73\times 10^{-12}\times 5.976\times 10^{24}}{8\times 10^6+6378\times 10^3}}$
This simplifies to:
$v_{Ac}=5266.43m/s=5.27\times 10^3m/s$
Now we determine the change in speed as
$v_p=\sqrt{\frac{2GM_er_a}{r_p(r_p+r_a)}}$
We plug in the known values to obtain:
$v_p=\sqrt{\frac{2\times 66.73\times 10^{-12}\times 5.976\times 10^{24}\times 25.378\times 10^6}{14.378\times 10^6(14.378\times 10^6+25.378\times 10^6)}}$
$\implies v_p=5950.58m/s$
We know that $v_p=v_{Ae}$
$\implies v_{Ae}=5950.58m/s$
Now $\Delta v=v_{Ae}-v_{Ac}$
We plug in the known values to obtain:
$\Delta v=5950.58-5266.43$
$\implies \Delta v=684m/s$