Q$\mathrm{.}1$

Engineers are considering whether or not to add a climbing lane to

a single carriageway road in order to allow overtaking of slower

moving vehicles. The gradient of the road is $+6\mathrm{.}5\%,$ and in order

to justify the addition of a climbing lane, the maximum speed of a

standard heavy goods vehicle must be less than $50$ km$/$h$.$

A standard heavy goods vehicle has a mass of $44,000$kg$,$ a frontal

area of $8\mathrm{.}5$m$2$ and a coefficient of drag $($CD$)$ of $0\mathrm{.}65.$ Air density is

$1\mathrm{.}3$ kg$/$m$3.$ The maximum power available to overcome

aerodynamic, rolling and grade resistance is $420$kW$.$

Calculate the maximum speed of the standard heavy goods

vehicle on this gradient using any suitable approach including trialand$-$error. Is a climbing lane justified?

$[8$ marks$]$