Alternative Homework Assignment: Tailgating
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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 behind the car ahead of you. Tailgating leads to multiple car crashes when one of the cars in a line suddenly slows down. The question we want to answer is: "How close is too close?"
To do this, let's suppose you are driving on the highway at a speed of 100 kilometers an hour (a bit more than 60 mph). The car ahead of you suddenly puts on its brakes. We need to calculate a number of things: how long it takes you to respond; how far you travel in that time; how far the other car traveled in that time.
(a) First let's estimate how long it takes you to respond. Two times are involved: how long it takes from the time you notice something happening till you start to move to the brake, and how long it takes to move your foot 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 middle of one of the short sides hanging straight down. Place your thumb and forefinger opposite the bottom of the paper. Have your friend release the paper suddenly and try to catch it with your thumb and forefinger. Measure how far the paper fell before you caught it. Do this three times and take the average distance. Assuming the paper was falling freely without air resistance (not bad for the paper falling sideways), calculate how much time it took before you caught it, t1.
Estimate the time (t2) it takes you to move your foot from the gas pedal to the brake pedal. Your reaction time is t1 + t2.
(b) If you brake hard and fast, you can bring a typical car to rest from 100 kph (about 60 mph) in 5 seconds. Calculate your acceleration, -a0, assuming that it is constant.
(c) Suppose the car ahead of you begins to brake with an acceleration -a0. 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 velocity over this time interval?)
(d) You see him start to slow immediately (an unreasonable but simplifying assumption). If you are also traveling 100 kph, how far (in meters) do you travel before you begin to brake? If you can also produce the acceleration -a0 when you brake, what will be the total distance you travel before you come to a stop?
(e) In the above calculations we have assumed that you were paying close attention 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 while you are singing a song, or in the middle of a conversation. Try catching the paper while distracted five times, and take the slowest of these times to be your worst case senario t1.
(f) Repeat steps (b-d) using your worst case senario t1 to come up with a worst case senario stoping distance.
(g) Discuss, on the basis of these calculations, what you think a safe distance is to stay behind a car at 60 mph. Would you include a safety factor beyond what you have calculated here? How much?
(h) It is sometimes difficult to estimate distances accurately when driving on the highway. An easy way to estimate the distance to the car ahead 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 corresponds to the distance you have calculated as being safe. How does this compare to the 2 second following distance recomended by traffic experts?
|Work supported in part by NSF grant DUE-9455561|
These problem 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 (AHA's) Problem sight is mentioned and the URL given.
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Comments and questions may be directed to E. F. Redish
Last modified June 25, 2002