Consider that a $70-$deg sphere cone planetary entry capsule with a nose radius $(R_n)$ of $0\mathrm{.}66$ m and $2\mathrm{.}65$ m maximum diameter is used for Mars entry $($similar to the Mars Pathfinder spacecraft$).$Assume that this system enters the Mars atmosphere with a mass $($m$)$ of $580$ kg at an entry velocity of $(V_0)7\mathrm{.}6$ km$/$s at an altitude of $(h_0)$ of $125$ km and with an entry flight path angle $()$ of $14\mathrm{.}2$ deg. Also assume a ballistic entry with constant flight path angle and constant drag coefficient of $($Cd$)1\mathrm{.}65.$ Use a density vs altitude model given by $(h)={}_0{e}^{-ah}$ where ${}_0=0\mathrm{.}057$ kg$/$m$^3$ and $a=1\mathrm{.}275{10}^{-4}{m}^{-1}$ in your calculations. Use a value of $3\mathrm{.}75\frac{m}{s^2}$ for the gravitational acceleration on Mars.

a$.$ For the given nominal conditions above: $($i$)$ Determine the ballistic coefficient for the probe. $($ii$)$ Determine the peak deceleration of the probe in terms of earth g$\text{'}$s$,$ the peak deceleration altitude, and the velocity at the peak deceleration conditions. $($iii$)$ Plot altitude vs$.$ velocity and altitude vs$.$ the deceleration $($in terms of earth g$\text{'}$s$)$ between the entry altitude and an altitude of $10$ km $.$