A PARTICULARLY FRIGHTENING experience when you are driving a car is making an emergency stop to avoid a collision. How should you apply the brakes in order to make the shortest or safest stop? Should you push the brake pedal all the way down and lock the wheels (that is, prevent them from rotating)? Or should you push it down hard but not quite hard enough to lock the wheels? Some driving instructors and a few physicists argue strongly for the first strategy, whereas some physics textbooks argue just as strongly for the second. Indeed, some modern cars have a computer-controlled brake system that automatically adjusts the brakes to keep the wheels from locking. Suppose your car lacks such a system. Which strategy should you follow?

To decide the matter you must consider the frictional force on the wheels from the road. When a wheel rotates smoothly without slipping, the friction is said to be static friction. The size of the frictional force matches the force the wheel exerts parallel to the road. When the car is stationary and exerts no force parallel to the road, there is no friction on a wheel. When the car accelerates and the engine attempts to turn the wheels faster, the driving wheels then push the road toward the rear and the road pushes them forward. It is this forward force on the wheels from the road that propels the car.

When the brakes retard the rotation of the wheels, the wheels push in the forward direction and experience a frictional force toward the rear, which is what slows the car. The upper limit to the force due to static friction is set by the product of the downward force on a wheel and a characteristic of the tire-road interaction called the coefficient of static friction. Normally the downward force in this equation is the amount of the car's weight supported by the wheel. The coefficient of static friction is largely a measure of the small-scale roughness of the tire and the road. A typical value for a dry asphalt road is .8.

If the force on a wheel from the road exceeds the upper limit of the static friction, the wheel begins to slip, and the frictional force on the wheel is then said to be sliding friction. If the brake is applied hard enough, it almost immediately locks the wheel. The frictional force points toward the rear of the car and its magnitude is equal to the product of the downward force on the wheel and a coefficient of sliding friction. Note that although the strength of the force due to static friction can vary from zero up to some maximum amount, sliding friction is set at the size given by the product. (Actually there can be a small variation in sliding friction owing to the speed at which the wheel slips over the road and other factors.)

The coefficient of sliding friction is less than the coefficient of static friction. For example, when a standard tire slides over dry asphalt, the coefficient of sliding friction may be .6. The reduction in friction when sliding begins results from several factors, but the most important one involves the heat generated by the sliding. When the road consists of bituminous materials such as asphalt, they melt and a skid mark is left. The tire may also melt. In any case the sliding produces a fluid layer between the wheel and the road that lubricates the motion and reduces the coefficient of friction.

And so should you lock the wheels in an emergency braking or not? If you brake hard without locking the wheels, the

static friction on the wheels can be as large as its upper limit. If instead you lock the wheels and they begin to slide, the friction is smaller than that upper limit because of the lubrication. Since in each case it is friction that brings the car to a stop, the best decision seems to be the first one, in which case the friction is larger and the car should stop in the shortest distance. Thus state some physics textbooks.

Driving experts sometimes differ with the conclusion. At least one has argued that when you are in an emergency braking situation, you surely do not have time to find the braking that maximizes the static friction while avoiding any locking of the wheels. The point is certainly correct: if you adjust and then readjust the braking, the time required must inevitably add to the stopping distance.

Suppose you ignore that practical point. How then do you answer the question about braking methods? The matter was taken up in 1979 by Daniel P. Whitmire and Timothy J. Alleman, who were then at the University of Southwestern Louisiana. They pointed out that, contrary to the standard textbook argument, experiments reveal that a stop in which all four wheels are locked usually requires less distance than a stop without any sliding. Their key point is that during braking, forces due to friction create torques on the car that significantly change the downward forces on the wheels and hence the frictional force.

To understand their point, consider a car that is moving forward when the brakes are applied [see illustration at right]. If none of the wheels slides, then each experiences a static frictional force toward the rear that slows the car. The forces also create torques that attempt to rotate the car vertically around its center of mass, so that the rear rides up and the front is pushed down. For each wheel the torque is equal to the product of the friction and a certain distance that is called a lever arm. To find the lever arm, mentally extend a frictional force under the car. The lever arm is a line perpendicular to the extension that passes through the car's center of mass. Since the friction is at road level, the length of a lever arm is the height of the center of mass. You can determine the direction of rotation associated with a torque by imagining that the force attempts to rotate the lever arm itself around the center of mass.

During hard braking the rotation of the car may be felt if the car's suspension system yields sufficiently: the car pitches forward, increasing the weight on the front wheels and decreasing the weight on the rear wheels. Even if the car were rigid, and so unable to rotate, the impulse to rotate would produce the same result: the torques would relieve some of the downward force on the rear wheels and increase the downward force on the front wheels. Hence the upper limit of the static friction becomes larger on the front wheels and smaller on the rear wheels than before the braking began.

To see how these changes figure in braking, Whitmire and Alleman considered several situations. For comparison they employed the textbook example in which the torques are not considered and each wheel is assumed to experience the same upper limit to the static friction. Let D be the stopping distance under these circumstances. Next they considered the action of the torques. Suppose that the brakes are identical and the driver has got the rear tires on the verge of slipping. The friction on the rear wheels is then at the upper limit for static friction, but the upper limit has been diminished by the torques. Since the braking on the front and the rear wheels is assumed to be identical, the front wheels must experience the same amount of friction. With the friction small all around, the stopping distance may be as large as 1.5 D.

Such controlled braking may actually be less effective at stopping a car than a full slide. The stopping distance for a fully sliding car is not altered by the torques, because the reduction of the friction on the rear wheels is exactly matched by the increase of the friction on the front wheels, and so the combined friction is the same as if there were no torques. As a result, if the coefficient of sliding friction is 20 percent less than the coefficient of static friction, the stopping distance for a fully sliding car is 1.25 D, which may be appreciably better than the distance for controlled braking.

The stopping distance for controlled braking can be reduced if the brakes are applied firmly enough so that the rear

wheels begin to slide and the front wheels are on the verge of sliding. Then the combination of the maximum static friction on the front wheels (which is large because of the torques) and the sliding friction on the rear wheels stops the car in a distance that is only slightly longer than D and hence is smaller than the stopping distance for a fully sliding car.

The brakes on cars are often biased to counteract the torques created by the friction: when you press down on the brake pedal, you engage the front brakes more than the rear brakes. Suppose the bias is so strong that the front wheels are put on the verge of sliding while the rear wheels are far from the sliding limit. Then the front wheels experience the maximum static friction while the rear wheels may experience only a small static friction, and again the stopping distance can be dangerously larger than D, and even larger than the distance for a fully sliding car.

The situation improves if the brakes are applied hard enough so that the front wheels begin to slide and the rear wheels are put on the verge of sliding. Then the front wheels experience sliding friction while the rear wheels experience the maximum static friction, and the stopping distance is only slightly larger than D.

There is one more condition to consider. The brake bias can be adjusted so that front and back wheels can be simultaneously put on the verge of sliding and all the wheels experience the maximum static friction. Only in this ideal case is the stopping distance D, as predicted by a textbook calculation. Such adjustment, however, may not be practical because it depends on the coefficient of static friction. Although the bias can be optimized for one type of road surface, it would then not be optimum for a different type of road surface with which the tires experience a different coefficient of static friction. (Here one sees the rationale for computer-controlled brake systems: the bias can be automatically adjusted for any road.)

In summary, then, locking all four wheels and sliding may be the best strategy in some emergency stops if the idea is to minimize the stopping distance. Its advantage is greatest if the torques on a particular car tend to put the rear wheels on the verge of sliding before the front wheels reach that point. The strategy does have one serious flaw, however: if the car is fully sliding, you have virtually no control over where it is headed, because of the lubricating fluid between the tire and the road. If the car happens to be turning when it begins to slide, you might quickly spin out of your lane of traffic. Spin can also be initiated if the wheels do not experience identical sliding friction or if the road is tilted or crowned, as it often is to improve drainage. A stop with the car both sliding and spinning might prove to be even more dangerous than a controlled stop, even if the controlled stop requires a greater distance.

The possibility of "spinout" was investigated in 1984 by William G. Unruh of the University of British Columbia for situations in which only the front wheels or only the rear wheels begin to slide. As you may have noticed in driving emergencies, if only the front wheels lock, the car is often stable in the sense that it continues to point forward, but if only the rear wheels lock, the car is likely to spin around until it points toward the rear and travels backward down the road.

To follow a simple version of Unruh's analysis (he also offered a highly refined version), consider a car that suddenly begins to spin counterclockwise [see illustration below]. Suppose that the front wheels are locked and the rear wheels are still rolling. Since the car is spinning, the rear wheels must be sliding sideways and experiencing a sliding frictional force that is parallel to the rear axle. The front wheels also experience sliding friction, but because they are locked, the forces point directly backward, opposite to the intended direction of travel.

Each frictional force generates a torque that tends to rotate the car horizontally about its center of mass To evaluate the torques, take an overhead view of the car and ignore the height of the car's center of mass above the road. Then, to find the lever arms associated with the torques, extend an arrow representing each force until it intersects a perpendicular line that goes through the car's center of mass. The length of the perpendicular line is the lever arm, and the direction of the rotation caused by a torque can be inferred by imagining that the force tends to rotate the lever arm itself about the center of mass.

When the front of the car has just begun to spin to the left, the approximately equal torques from the front wheels attempt rotations in opposite directions and cancel, but the torques from the rear wheels both promote a clockwise rotation that opposes the spin. Even if the car turns appreciably, so that the torque from the left front wheel increases because of a lengthened lever arm, the torques due to the friction on the rear wheels continue to dominate because of their longer lever arms, and eventually the spin is halted and then reversed until the car is again pointed in the proper direction. Hence when only the front wheels lock, any chance rotation of the car is automatically corrected by the friction on the rear wheels.

Suppose instead that only the rear wheels are locked. They experience frictional forces that are directed backward, while the still rolling front wheels experience frictional forces that are parallel to the front axle [see Figure 5]. Early in the spin the torques from the rear wheels essentially cancel, because they attempt rotation in opposite directions. The torques from the front wheels do not cancel, because they both promote a counterclockwise rotation, and so the car continues to spin. As the car's misalignment increases, the torque from the right rear wheel increases because of its lengthened lever arm, but it is still no match for the combined torques from the front wheels. And so the spin goes unchecked and the car turns completely around until it is traveling down the road backward.

Unruh also considered how the position of a car's center of mass influences the likelihood of such a reversal. If the engine is in the front of the car, the center of mass is forward of the car's midpoint and less weight rides on the rear wheels than on the front ones. The rear wheels are then more likely to lock first during an emergency stop, sending the car into a spin.

When the engine is in the back of the car, as it was in a Volkswagen "Beetle" once owned by Unruh, the car is much stabler against spin. Then the center of mass is behind the car's midpoint and more weight rides on the rear wheels than on the front ones. The front wheels are more likely to lock first, and so any spin is quickly corrected by the friction on the rear wheels. Some drivers arrange for such stability even in a car where the engine is mounted in the front-they pack bags of sand or some other heavy objects in the back of the car. The precaution might indeed help a driver who faces the possibility of braking on an icy street where otherwise the rear wheels would easily become locked.

Two initial conditions affect the possibility that a car with a front-mounted engine will spin when its rear wheels are locked. If the car is suddenly at an angle to the intended direction of travel but has no spin yet, it will begin to spin out of control if the angle is larger than some critical value. The critical value is small for most street speeds, and it gets even smaller for progressively higher speeds. If instead the car is initially pointed in the intended direction and is given some spin by, say, a nonuniform part of the road, it will continue to spin if the initial spin is larger than some small critical value. Again speed works against the driver. In most practical situations, particularly at high speed, the locking of the rear wheels is certain to send the car into an uncontrollable spin.

The existence of a critical angle and a critical initial spin has to do with the fact that the frictional forces not only create torques on the car but also slow it and tend to change its direction of travel. If neither the critical angle nor the critical initial spin has been exceeded, the action of the forces actually stabilizes the car. For example, if the car is suddenly turned slightly leftward, the friction that then works on the front wheels might accelerate the car's center of mass to the left, thereby bringing its direction of travel in line with the direction in which it is pointing before the spin can build.

You might be able to correct a spin by turning the front wheels toward the intended direction of travel; at least, that is what is commonly advised. If the speed is low, the scheme should work even if you turn the wheels somewhat too far and then have to turn them back. If the speed is high, however, any error on your first try will shoot the car through the proper orientation and spin it out of control in the opposite direction.

All the braking schemes I have presented ignore a practical aspect in an emergency: reaction time. Since you cannot recognize a dangerous situation and apply the brakes instantaneously, you must necessarily travel down the road a certain distance before the braking begins. Some legal textbooks that deal with traffic accidents place lower limits on reaction times. If the danger is recognized quickly and calls for nothing more complex than braking, you might need only a quarter of a second to perceive the danger and then another quarter of a second to apply the brake. Suppose that you are traveling at 90 kilometers per hour (about 55 miles per hour) when the emergency arises. Then the minimum distance based on your reaction to the danger and full application of the brake is about 13 meters. Of course, if the danger is difficult to ascertain or if you are distracted momentarily or slowed by alcohol, the distance associated with your reaction can be much longer.

There is another factor in an emergency stop-one that you are pretty well stuck with. It is the coefficient of friction, either static or sliding, between the wheels and the roadway. Extensive experiments have shown that the coefficient can vary considerably depending on the type of tire and the degree to which it is inflated. The coefficient can also vary with the type of paving material, how long the pavement has been in place and how heavily traveled the road has been. For example, a concrete road polished by heavy use can have a coefficient of sliding friction that is only 70 percent of what it was when the surface was new, and of course the decreased coefficient considerably lengthens the minimum stopping distance.

Bibliography

THE PHYSICS OF TRAFFIC ACCIDENTS. Peter Knight in Physics Education, Vol.10, No. 1, pages 30-35; January, 1975.

EFFECT OF WEIGHT TRANSFER ON A VEHICLE'S STOPPING DISTANCE. Daniel P. Whitmire and Timothy J. Alleman in American Journal of Physics, Vol. 47, No. 1, pages 89-92; January, 1979.

AUTOMOBILE STOPPING DISTANCES. L. J. Logue in The Physics Teacher, Vol.17, No. 5, pages 318-320; May, 1979.

INSTABILITY IN AUTOMOBILE BRAKING. W. G Unruh in American Journal of Physics, Vol. 52, No. 10, pages 903-909, October, 1984.

TRAFFIC ACCIDENT INVESTIGATOR'S MANUAL FOR POLICE. J. Stannard Baker and Lynn B. Fricke. The Traffic Institute, Northwestern University, 1986.

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