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!headlights.jpg|width=650!
{excerpt:hidden=true}How long can you drive at constant velocity before you have to hit the brakes, assuming standard night detection distances?{excerpt}
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|http://onlinepubs.trb.org/Onlinepubs/nchrp/nchrp_rpt_400.pdf]) 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?
h4. Solution
{toggle-cloak:id=sys} *System:* {cloak:id=sys}Your car will be treated as a [point particle].{cloak}
{toggle-cloak:id=int} *Interactions:* {cloak:id=int}The assumption of constant [velocity] implies that there is no _net_ [interaction] applied to the [system].{cloak}
{toggle-cloak:id=mod} *Model:* {cloak:id=mod}[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.{cloak}
{toggle-cloak:id=app} *Approach:*
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{toggle-cloak:id=diag} {color:red} *Diagrammatic Representation* {color}
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Our [model] has only one relevant [Law of Change], namely:
{latex}\begin{large} \[ x = x_{\rm i} + vt\] \end{large}{latex}
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|position], 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.
!headlights2.png|width=650!
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.
{note}In any problem involving vectors, you will be required to clearly define a coordinate system if the problem does not specify one.{note}
To illustrate the importance of a complete [coordinate system], consider these two possible correct [coordinate systems|coordinate system] for the problem.
!headlights3.png!
Notice that because we have measured the distances from the deer, we either have to make the car's [positions|position] negative, or else we must assign the car a negative [velocity]. We will use the first version, with negative positions and a positive [velocity].
{info}It is possible to shift the position axis so that the driver first sees the deer at _x_~i~ = 0. Then, the car can have both a positive final _x_ position and also a positive velocity. You must perform the shift carefully, however. In Whatthe shifted coordinates, what is the position at which the car hits the brakes?{info}
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{toggle-cloak:id=math} {color:red} *Mathematical Representation* {color}
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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.
{warning}The terms "final" and "initial" in physics equations are often misleading. These refer to arbitrarily chosen points in the problem. Often, they do not correspond to the absolute final and absolute initial parts of the motion. There are only two requirements that must be satisfied when choosing these points: (1.) the final point must be reached later in time than the initial point and (2.) the equation you are using must accurately describe the motion at all times in between the chosen points. (Of course, it is also of practical importance that you know something about the motion at the points you choose.){warning}
Solving the equation for time and inserting the appropriate numbers gives:
{latex}\begin{large} \[ t = \frac{x-x_{\rm i}}{v} = \frac{-43\:{\rm m} - (- 95\:{\rm m})}{27.8\:{\rm m/s}} = 1.9\:{\rm s}\] \end{large}{latex}
{note}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\\
{latex}\begin{large}\[ \frac{100\:{\rm km}}{\rm h} \times \frac{1\:{\rm h}}{60\:{\rm min}}\times\frac{1\:{\rm min}}{60\:{\rm s}}\times\frac{1000\:{\rm m}}{1\:{\rm km}}=27.8\:{\rm m/s}\]\end{large}{latex}
{note}
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)?
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