Overdriving Headlights

According to a study performed in 1997 by Texas A&M University for the Transportation Research Board of the National Research Council (available at their website) people driving at night using low-beam headlights lose their ability to see a large animal like a deer at a distance of anywhere from about 80 m to 100 m (see the figure above taken from NCHRP Report #400). Another study performed in 1998 by the U.S. Army for the Department of Transportation's National Highway Traffic Safety Administration indicated that 43 m is a good braking distance for a sedan with anti-lock brakes traveling at 100 km/h and 50 m is a poor braking distance. If you were driving down the highway at 100 km/h at night using your low-beam headlights, how much time do you have to react to an animal in the road, assuming you notice it at a distance of 95 m and your car has a braking distance of 43 m? In other words, how much time can elapse between the instant you first spot the animal and the instant you hit the brakes without resulting in an accident?

Solution

System:

Your car will be treated as a point particle.

Interactions:

The assumption of constant velocity implies that there is no net interaction applied to the system.

Model:

1-D Motion (Constant Velocity). Although the car will eventually stop, we are interested in the time that elapses from the instant the animal is recognized until the instant the brakes are pressed. During that time interval, the car is coasting at 100 km/h.

Approach:

Diagrammatic Representation

Our model has only one Law of Change, namely:

To use it properly in this case, however, requires consideration of the meaning of the givens. We are looking for the time, and the magnitude of the velocity is clearly 100 km/h, but we run into some trouble with the position values. The equation only involves two positions, but we are given five quantities with units of meters.

To determine which positions to use, it is helpful to sketch the situation. The important information in the problem statement can be summarized as shown below.

Quickly sketching the problem has immediately told us that the relevant positions will be 43 m and 95 m. Before plugging into the equation, however, we must consider one more detail. The sketch made above was not complete. We have given the distances to the deer, but we have not set up a true coordinate system. Failing to set up a careful coordinate system can result in important sign errors, as it would here.

To illustrate the importance of a complete coordinate system, consider these two possible correct coordinate systems for the problem.

Notice that because we have measured the distances from the deer, we either have to make the car's positions negative, or else we must assign the car a negative velocity. We will use the first version, with negative positions and a positive velocity.

Mathematical Representation

Now we want to plug the appropriate numbers into the equation. Note that even though the car finally comes to rest at x=0, we must use x = – 43 m as the final position in the equation for constant velocity. The reason is that x= – 43 m is where the car hits the brakes. Once the brakes are engaged, our model no longer applies. The car's velocity will not be constant. Thus, for the part of the problem that is modeled as constant velocity, the final position will be x = – 43 m.

Solving the equation for time and inserting the appropriate numbers gives:

Don't forget to convert to a consistent set of units. In the equation above, we converted 100 km/h into m/s by using

NCHRP Report #400, using studies reported in the literature, reports an approximate average reaction time to an unexpected object of 1.3 seconds. In this case, then, we would say that the driver has enough time to react to the deer and stop the car. Would that still be true (again assuming a speed of 100 km/h) if the driver was driving a car with a stopping distance of 50 m and a spotting distance of 80 m (both within the range of the studies cited above)?