$A$ racing car travels on a track (without banking) $ABCDEFA$. $ABC$ is a circular arc of radius $2R$. $CD$ and $FA$ are straight paths of length $R$ and $DEF$ is a circular arc of radius $R = 100 \, m$. The coefficient of friction on the road is $\mu = 0.1$. The maximum speed of the car on straight paths is $50 \, m/s$. Find the minimum time for completing one round.

(D) The frictional force provides the necessary centripetal force for circular motion.
For a circular path of radius $r$,the maximum speed $v$ is given by $\frac{mv^2}{r} = \mu mg$,which simplifies to $v = \sqrt{\mu rg}$.
$1$. For path $ABC$ (radius $2R = 200 \, m$):
Length $L_1 = \frac{3}{4} \times (2\pi \times 2R) = 3\pi R = 300\pi \, m$.
Speed $v_1 = \sqrt{0.1 \times 200 \times 10} = \sqrt{200} \approx 14.14 \, m/s$.
Time $t_1 = \frac{300\pi}{14.14} \approx 66.66 \, s$.
$2$. For path $DEF$ (radius $R = 100 \, m$):
Length $L_2 = \frac{1}{4} \times (2\pi R) = \frac{\pi R}{2} = 50\pi \, m$.
Speed $v_2 = \sqrt{0.1 \times 100 \times 10} = 10 \, m/s$.
Time $t_2 = \frac{50\pi}{10} = 5\pi \approx 15.71 \, s$.
$3$. For straight paths $CD$ and $FA$ (length $R = 100 \, m$ each):
Total length $L_3 = R + R = 200 \, m$.
Speed $v_3 = 50 \, m/s$.
Time $t_3 = \frac{200}{50} = 4.0 \, s$.
Total time $T = t_1 + t_2 + t_3 = 66.66 + 15.71 + 4.0 = 86.37 \, s$.