The $2024$ MAZDA MX$-5$ Miata is shown in Figure $2.$ It has a mass of $1118$ kg and a mass moment of inertia about the yaw axis of approximately $1797kg{m}^2.$ The wheelbase is $2\mathrm{.}309$ m $.$ The centre of mass is such that $50\%$ of the static load is on each of the front and rear axles. The steering gear ratio of the power$-$assisted steering is $15$:$1.$ The cornering stiffness of each of the front and rear tires is $19\mathrm{.}5k\frac{N}{r}ad$ and $22\mathrm{.}0k\frac{N}{r}ad,$ respectively. The centre of mass is $53$ cm above the ground. Consider the case where the vehicle is travelling at a

Figure $2$: Mazda Miata.

speed of $100k\frac{m}{h}$ while negotiating a constant$-$radius turn resulting in a lateral acceleration of $0\mathrm{.}4$ g $.$

$($a$)$ Determine the radius of the steady$-$state turn.

$($b$)$ Determine the vehicle's yaw rate and the curvature of the vehicle's trajectory.

$($c$)$ Determine the steering wheel angle required to negotiate the turn.

$($d$)$ Determine the front and rear cornering forces and slip angles.

$($e$)$ Determine either the vehicle's characteristic speed or critical speed, as appropriate, during the turn; and indicate the significance of the calculated value.

$($f$)$ Estimate the rise time of the yaw rate as the vehicle transitioned from rectilinear motion into the steady$-$state turn assuming the vehicle entered the turn with the constant forward speed of $100k\frac{m}{h}.$