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...

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 answer this question, 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 ruler to do this. Take the ruler and have a friend hold it from the one end hanging straight down. Place your thumb and forefinger opposite the bottom of the ruler. Have your friend release the ruler suddenly and try to catch it with your thumb and forefinger. Measure how far it fell (in meters) before you caught it. Do this three times and take the average distance -- we will call this r. Assuming the ruler was falling freely without air resistance, you can calculate how much time it took before you caught it, t1, using the equation r = (5 m/s2) * t12.

Also, you can estimate the time, t2, it takes you to move your foot from the gas pedal to the brake pedal. Your total reaction time is t1 + t2.

What was your t1?

1 s

What was your estimate for t2?

2 s

So what is your reaction time? (Use your estimates.)

3 s

B. If you brake hard and fast, you can bring a typical car to rest from 100 kph (about 60 mph) in 5 seconds.

B.1 Calculate the magnitude of your acceleration, −a0, assuming that it is constant.

4 m/s2

Why did we put a minus sign in front?

5This answer has not been graded yet.

B.2 Suppose the car ahead of you (which was also going 100 kph) begins to brake with an acceleration −a0 from B1. How far will he travel before he comes to a stop? (Hint: How much time will it take him to stop?)

6 m

B.3 What will be his average velocity over this time interval?

7 m/s

C. Now we can put these results together into a semi-realistic situation. You are driving on the highway at 100 km/hr and there is a car in front of you going at the same speed.

C.1 You see him start to brake immediately. (An unreasonable but temporarily useful simplifying assumption.) If you are also traveling 100 kph, how far (in meters) do you travel before you begin to brake, using your reaction time from part A.

Minimum distance = 8 m

If you can also produce the acceleration −a0 from part B1 when you brake, what will be the total distance you travel before you come to a stop?

9 m

C.2 If you don't notice the car ahead of you beginning to brake for 1 second, how much additional distance will you travel?

10 m

C.3 On the basis of these calculations, what do you think is a safe distance to stay behind a car at 60 mph? Express your distance in "car lengths" (about 15 feet).

11 car lengths