$3.$ Powertrain$-$Limited Acceleration Analysis $($Rough$).$ Consider a $\backslash (3000-\backslash$mathrm$\{$lb$\}\backslash )$ car with an engine operating at about constant power of $150$ hp $,$ a smoothly operating drive$-$train about $\backslash (85\backslash \%\backslash )$ efficient, and a mass factor around $1\mathrm{.}3$ in $\backslash (1^\{\backslash$text $\{$st $\}\}\backslash )$ gear $($up to $25$ mph $),$ and $1\mathrm{.}1$ for higher gears $($above $25$ mph $).$

$($a$)$ Plot the acceleration versus velocity $($up to $60$ mph $),$ and the distance traveled versus time $($also until $60$ mph is reached$),$ assuming power$-$limited acceleration

$($Hints: The mass factor is an amplification of the vehicle mass to account for the effects of rotary inertia of drive train components. The efficiency here represents the ratio of the power at the tires to that from the engine.$)$

$($b$)$ Repeat for the case in which the tire$-$road coefficient in icy conditions $($now allowing for traction$-$limited acceleration$),$ but with a little gravel dispensed onto the road, is around $0\mathrm{.}3.($Ignore load transfer; treat also as if all$-$wheel drive, i$.$e$.,$ "drive axle load" same as full vehicle weight.$)$