$[12$ points$]$ On Mars, the acceleration due to gravity is about $3\mathrm{.}7\frac{m}{se{c}^2}($compared to $g$~~$9\mathrm{.}8\frac{m}{se{c}^2}$ on earth$).$ Suppose you throw a rock on Mars in such a way that its velocity after $2$ seconds is $2\mathrm{.}6\frac{m}{s}$ and its height after $2$ seconds is $17\mathrm{.}6$ m $.$

a$)$ Show that the rock's velocity $t$ seconds after being thrown is given by $v(t)=-3\mathrm{.}7t+10.$ That is$,$ start with the acceleration formula $a=-3\mathrm{.}7$ and do some calculus to arrive at an equation for the rock's velocity. Be sure to include a $"+c"$ along the way and then find the value of $c.$

b$)$ Show that the height of the rock $t$ seconds after being thrown is given by $h(t)=-1\mathrm{.}85t^2+10t+5.$ Be sure to include $a"+$c $"$ along the way and then find the value of c $.$ The final formula should have the letters $h$ and $t$ in it but no other letters! Hint: Start with the formula for the rock's velocity from part $($a$).$

c$)$ How long does the rock take to hit the ground? Round your answer to the nearest hundredth and include units.

d$)$ What is the rock's velocity when it hits the ground? Round your answer to the nearest hundredth and include units.