Problem $3.$ Derive the braking$-$distance formula

$D_b=\frac{u_0^2}{2g(f+G)}$

using a free$-$body diagram, where $u_0$ is the initial velocity, $g$ is acceleration due to gravity, $f$ is the coefficient

of braking friction, and $G$ is the percent grade $($uphill is positive$).$ Assume that the vehicle's brakes are

powerful enough that the limiting factor is friction between the tires and road. Remember that $D_b$ is

measured horizontally, not along the angle of the roadway $($that is$,D_b$ is the distance between the initial

and stopping points as measured from an aerial photo$),$ and that the grade $G=tan,$ where $$ is the angle

between the horizontal and the roadway surface in the direction of travel.