Answer
$Altitude=17118.6\space km$
$Speed= 1.448\space km/s$
Work Step by Step
(b) Here we use the equation $T=\sqrt {\frac{4\pi^{2}r^{3}}{GM_{M}}}$ where, T - Orbital period, V - speed of the spacecraft, $M_{M}$ - mass of the Mars, G - universal gravitational constant, r - radius of the orbit.
$T=\sqrt {\frac{4\pi^{2}r^{3}}{GM_{M}}}$ ; Let's plug known values into this equation.
$T= 2\pi \sqrt {\frac{r^{3}}{GM_{M}}}$
$1.03\times24\times3600\space s=2\pi \sqrt {\frac{r^{3}}{6.67\times10^{-11}\space Nm^{2}/kg^{2}\times0.642\times10^{24}\space kg}}$
$14163.5\space s= \sqrt {\frac{r^{3}}{4.3\times10^{13}m^{3}/s^{2}}}$
$14163.5\space s\times\sqrt {4.3\times10^{13}m^{3}/s^{2}}=\sqrt {r^{3}}$
$8.6\times10^{21}m^{3}=r^{3}$
$20508.6\space km=r$
Altitude (h) = r - radius of the mars = (20508.6-3390) km
h = 17118.6 km
(a) Here we use the equation $V=\sqrt {\frac{GM_{M}}{r}}$ to find the speed of the spacecraft.
$V=\sqrt {\frac{GM_{M}}{r}}$; Let's plug known values into this equation
$V=\sqrt {\frac{6.67\times10^{-11}\space Nm^{2}/kg^{2}\times0.642\times10^{24}\space kg}{20508.6\space \times10^{3}m}}$
$V=\sqrt {\frac{4.3\times10^{13}}{20.51\times10^{6}}}\space m/s= 1148 \space m/s=1.448\space km/s$
