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In this problem we analyze the phenomenon of "tailgating"in a car on a highway at high speeds. This means traveling too close behindthe car ahead of you. Tailgating leads to multiple car crashes when oneof the cars in a line suddenly slows down. The question we want to answeris: "How close is too close?"

To do this, let's suppose you are driving on the highway at a speedof 100 kilometers an hour (a bit more than 60 mph). The car ahead of yousuddenly puts on its brakes. We need to calculate a number of things: howlong it takes you to respond; how far you travel in that time; how farthe other car traveled in that time.

(a) First let's estimate how long it takes you to respond. Two timesare involved: how long it takes from the time you notice something happeningtill you start to move to the brake, and how long it takes to move yourfoot to the brake. You will need a piece of paper (8.5x11), a meter stick,and a coin to do this.

Take a piece of paper and have a friend hold the paper from the middleof one of the short sides hanging straight down. Place your thumb and forefingeropposite the bottom of the paper. Have your friend release the paper suddenlyand try to catch it with your thumb and forefinger. Measure how far thepaper fell before you caught it. Do this three times and take the averagedistance. Assuming the paper was falling freely without air resistance(not bad for the paper falling sideways), calculate how much time it tookbefore you caught it, t_{1}.

Estimate the time (t_{2}) it takes you to move your foot fromthe gas pedal to the brake pedal. Your reaction time is t_{1} +t_{2}.

(b) If you brake hard and fast, you can bring a typical car to restfrom 100 kph (about 60 mph) in 5 seconds. Calculate your acceleration,-a_{0}, assuming that it is constant.

(c) Suppose the car ahead of you begins to brake with an acceleration-a_{0}. How far will he travel before he comes to a stop? (Hint:How much time will it take him to stop? What will be his average velocityover this time interval?)

(d) You see him start to slow immediately (an unreasonable but simplifyingassumption). If you are also traveling 100 kph, how far (in meters) doyou travel before you begin to brake? If you can also produce the acceleration-a_{0} when you brake, what will be the total distance you travelbefore you come to a stop?

(e) In the above calculations we have assumed that you were paying closeattention to the car in front of you and were anticipating a need to stop.Realistically, this is not always the case. To account for this, do part(a) again, but have your friend distract you. Try catching the paper whileyou are singing a song, or in the middle of a conversation. Try catchingthe paper while distracted five times, and take the slowest of these timesto be your worst case senario t_{1}.

(f) Repeat steps (b-d) using your worst case senario t_{1} tocome up with a worst case senario stoping distance.

(g) Discuss, on the basis of these calculations, what you think a safedistance is to stay behind a car at 60 mph. Would you include a safetyfactor beyond what you have calculated here? How much?

(h) It is sometimes difficult to estimate distances accurately whendriving on the highway. An easy way to estimate the distance to the carahead of you is to measure the time it takes your to reach a fixed object(such as a pavement marking) after the car ahead of you has passed it.As a final calculation, find the time interval at 100 kph that correspondsto the distance you have calculated as being safe. How does this compareto the 2 second following distance recomended by traffic experts?

Work supported in part by NSF grant DUE-9455561 |

These problems written and collected by K. Vick, E. Redish, and P. Cooney.
These problems may be freely used in classrooms. They may be copied and
cited in published work if *the Activity-Based Physics* (ABP) *Alternative
Homework Assignments* are mentioned and the source cited.

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Page last modified October 27, 2002