Vehicle at the Top of a Convex Over-Bridge

A vehicle crossing the crest of a convex (arched, hump-shaped) over-bridge is, for a brief instant, undergoing a small piece of vertical circular motion, with the bridge's radius of curvature r playing the role of the circle's radius. At the very TOPMOST point of the arch, exactly two forces act on the vehicle, both along the same vertical line: weight mg, downward, and the normal reaction N from the road, upward. Since this topmost point is the highest point of the (locally circular) path, the centre of curvature lies BELOW the road at that point, so the net downward force -- weight minus normal reaction -- must supply the centripetal force: $mg-N=\frac{mv^2}{r}$ As the vehicle's speed v increases, the required centripetal force $\frac{mv^2}{r}$ increases, and since mg is fixed, N must correspondingly DECREASE to make up the difference. Since the normal reaction is precisely the physical indicator of contact between the vehicle and the road, the vehicle stays on the road only as long as $N\geq0$; the borderline case $N=0$ (just losing contact) sets the upper speed limit $v_{max}=\sqrt{rg}$ Driving faster than this over the crest of the bridge means the vehicle would need a downward force greater than its own weight to stay on the curved path -- since the road ca …