$(15$ points$)$ You are participating in an electric vehicle racing competition, using remote$-$controlled toy cars like

the one shown in the figure above. According to the competition rules, the DC motor whose specs are given in

the table below must be selected as the traction motor, and it must be powered by a $3$ V battery. You design the

drivetrain of the vehicle such that one full rotation of the motor shaft corresponds to two full rotations of the

vehicle's traction $($rear$)$ wheels of radius $R=30mm.$

$3\mathrm{.}1$ You directly connect the DC$-$motor$-$controlled car to the battery and measure the maximum rotational

speed of its rear wheels as $3800$ rpm $($with the wheels not in contact with the ground$).$ Calculate the

friction torque $(T_f)$ in the vehicle's drive train in mNm $.$

$3\mathrm{.}2$ When the vehicle is moving on the ground at constant speed, the total traction force $(F_{tt})$ on both rear

wheels is given in mN by

$F_{tr}=A+B{v}^2$

where $v$ is the vehicle's speed in $\frac{m}{s},A=300mN$ and $B=5\mathrm{.}4\frac{g}{m}.$ Determine the maximum constant

speed $(v_{\text{m}\text{a}\text{x}})$ of the vehicle in $\frac{m}{s}$ when it is moving on the ground. Ignore the motion of the front

wheels; assume that the rear wheels roll without slipping and that the torque on the motor shaft is twice

that on the traction wheels.

$3\mathrm{.}3$ What is the actual efficiency $({}_{\text{a}\text{c}\text{t}})$ of the DC motor $($combined with the drivetrain$)$ at the $v_{\text{m}\text{a}\text{x}}$ value

calculated above?