Problem $1$: $(14$ points$)$

A car with mass of $1500$ kg is traveling at $80\frac{km}{h}.$ The brakes are fully applied at this moment, resulting in wheel skidding. The road conditions are for two scenarios: $($a$)$ on dry pavement with a coefficient of kinetic friction ${}_k=0\mathrm{.}8$; and $($b$)$ on icy road with a coefficient of kinetic friction ${}_k=0\mathrm{.}1.$ Assume gravity $g=9\mathrm{.}81\frac{m}{s^2}.$ Show all work.

a$)$ Draw the appropriate free body and kinetic diagram$($s$).(2$ points$)$

b$)$ Write the equations of motion based on the free body and kinetic diagram$($s$)$ written above. $(2$ points$)$

c$)$ Compute the expression of deceleration, $a,$ in terms of ${}_k.(2$ points$)$

d$)$ Determine the magnitude of deceleration on dry pavement, $a_{\text{d}\text{r}\text{y}}.(2$ points$)$

e$)$ Determine the stopping time on dry pavement, $t_{\text{d}\text{r}\text{y}}.(2$ points$)$

f$)$ Determine the magnitude of deceleration on icy road, $a_{\text{i}\text{c}\text{y}}.(2$ points$)$

g$)$ Determine the stopping time on icy road, $t_{\text{i}\text{c}\text{y}\text{.}}.(2$ points$)$