When you are driving on the freeway and following the car in front of you, how close is too close? Let's do an estimation. 1. Pick a car model (preferably the one you drive, but can also be any car of your dream), and find its stopping distance at highway speeds (you can usually find this type of data online).
2. Assuming that the car in front of you suddenly does a hard brake. For simplicity, assume that its braking performance is about the same as yours. Then also assume a reasonable amount of reaction time on your part (the time delay between seeing the brake lights lit up and applying your own brake). In order for you not to run into the car your are following, what's the closest distance you need to keep between the two cars?
3. Redo the same calculation if the vehicle in front of you is a typical big-rig truck. Find its braking data online.
4. There is a rule of thumb which says that you must stay one car length behind the car in front of you for every 10 mi/h of driving speed. From your calculation, does this rule make sense?

1) v= 90km/h  d = 70 m,  2)  x₁ = v t_r,  x₁ = 6.25 m, 3) x₁=6.25 no change

4) x = 22 m

Explanation:

1) for the first part, you are asked to find the minimum safety distance with the vehicle in front

The internet is searched for the stopping distance for two typical speeds on the highway

v (km/ h)      v (m/s)      d (m)

90                  25           70

100                 27.78      84

the safe distance is this distance plus the distance traveled during the person's reaction time, which can be calculated with infirm movement

              v = x / t_r

               x₁ = v t_r

the average reaction time is t_r = 0.25s for a visual stimulus and t_r 0.17 for an auditory stimulus

therefore the safe distance is

              x_total = x₁ + d

2) The distance is the sum of the distance traveled in the reaction

              x₁ = v t_r

for v = 90 km / h

              x₁ = 25 0.25

              x₁ = 6.25 m

for v = 100 km / h

              x₁ = 27.78 0.25

              x₁ = 6.95 m

the total distance is

               x_total = x₁ + d

for v = 90 km / h

             x_total = 25 0.25 + 70

             x_total = 76.25 m

this is the distance until the cars stop and do not collide

3) the stopping distance of a truck is

   v = 90 km / h       d = 100 m

in this case we see that the braking distance is much higher,

the safe distance is given by the distance traveled during the reaction, as the truck brakes slower than the car this distance does not change

4) let's analyze the empirical rule: maintain the length of a car for each increase in speed of v = 10 m / h = 4.47 m / s

for the car case at v = 90km / h = 25 m / s

according to this rule we must this to

           x = 25 / 4.47 = 5.6 cars

each modern car is about 4 m long so the distance is

           x = 22 m

we see that this distance is much greater than the reaction  distance so it does not make much sense

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