My wife and I were traveling in the mountains during a strong rainstorm. At one location, the rain had almost covered the road with dirt and gravel washed down from the uphill side. To continue required passage on the edge of the downhill side where a torrent of water was washing across the road and down the steep slope below. It was a harrowing experience since the force of the water was almost great enough to wash a car down the side of the mountain. A lighter car in front of us barely made it through.
This experience reminded me of a recent case where a young woman was driving through flood waters crossing over a road. The force of the water carried her car off the roadway into a creek channel. It is interesting to consider what happens in such a case.
As moving water rushes against the side of a car, it causes a transverse pressure on the car in the downstream direction. This pressure acts on the projected cross sectional area A of the part of the car that is under water, producing a force F in the transverse direction. The force is approximated by the equation:
F=0.5AnV2=Fw +F c
n is the density of water (n=1.9451 slug/ft 3) and V is the speed of the water. The pressure first acts on the tires, creating a force F w on the cross-sectional area of the tires below the bottom of the car. As the water rises above the bottom of the car body a force F c is exerted on the side of the car body. F is opposed by the force of friction F f, for which the maximum value is:
Where mu is the coefficient of friction, N is the force between the tires and the pavement, W is the loaded weight of the car, and B is the force of buoyancy of the car. B is equal to the weight of the water displaced by the car and is given by:
where L=length and C=width of the car, D is the depth of the water above the bottom of the car body, and g is gravitational acceleration. Note that A=LD.
Prior to slipping sideways F f=F. When F becomes greater than the maximum friction force the car begins to slip sideways. Setting F equal to the maximum value of F f and solving for D determines the depth at which slipping occurs.
If the depth of the water is deeper than D, the car will slip sideways or float on the water. The depth at which the car will begin to float is determined by setting B= W, but ifV>0 it will always begin to slide before it begins to float.
For the case involving a 2000 Toyota Avalon with a 165 lb passenger, W=3,603, L=192 inches, C=72 inches, and the radius of the tires is 13 inches. Using a typical downstream flow, V=10 mph, and for the road, mu=0.5. This