The graph below shows a model for the resultant horizontal force on a car, which varies as it accelerates from rest for $20$ seconds. The mass of the car is $1200$ kg $.$

$($a$)$ The acceleration of the car at time $\backslash ($ t $\backslash )$ seconds is a $\backslash (\backslash$mathrm$\{$m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash ).$ Show that $\backslash ($ a$=\backslash$frac$\{2\}\{3\}+\backslash$frac$\{$t$\}\{20\}\backslash ),$ for $\backslash (0\backslash$leq $\backslash$mathrm$\{$t$\}\backslash$leq $20\backslash )$

$($b$)$ Find an expression for the velocity of the car at time $\backslash ($ t $\backslash ).$

$($c$)$ Find the distance travelled by the car in the $20$ seconds.

$($d$)$ An alternative model assumes that the resultant force increase uniformly from $900$ to $2100$ newton during the $20$ seconds. Which term in your expression for the velocity would change as a result of the modification? Explain why.