## College Physics (4th Edition)

The safest speed is $7.85~m/s$
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 $7.85~m/s$