Answer
The safest speed is $7.85~m/s$
Work Step by Step
To find the safest speed, we can assume that the frictional force is zero.
Let $F_N$ be the normal force exerted by the road on the car. We can make an equation with the vertical components of the forces acting on the car:
$\sum F_y = 0$
$F_N~cos~\theta -mg = 0$
$F_N = \frac{mg}{cos~\theta}$
The horizontal component of the normal force provides the centripetal force to keep the car moving around the curve:
$F_N~sin~\theta = m~a_r$
$(\frac{mg}{cos~\theta})~sin~\theta = m~\frac{v^2}{r}$
$g~tan~\theta = \frac{v^2}{r}$
$v^2 = g~r~tan~\theta$
$v = \sqrt{g~r~tan~\theta}$
$v = \sqrt{(9.80~m/s^2)(120~m)~tan~(3.0^{\circ})}$
$v = 7.85~m/s$
The safest speed is $7.85~m/s$
