A graphical analysis of vehicle emergency stopping, including speeds at various times, speeds at various distances, and a rationale for the two-second rule.
Your stopping distance is the sum of two components, your Thinking Distance and the Vehicle Braking Distance.
Thinking Distance
Your thinking distance is the distance that your vehicle will travel
from the time that you first see the hazard
to the time that you apply the brakes.
This includes the time it takes you to recognise the danger,
to decide what avoiding action needs to be taken,
and to move your foot from the accelerator pedal
to the brake pedal;
this time is therefore longer than your reaction time alone.
Your thinking time is about two thirds of a second,
and during it your vehicle is still travelling at your initial speed.
Your thinking time does not vary with speed,
but the higher your speed the greater the distance you will travel during this time.
If you are tired after a long day, a long drive, or a large meal,
then your thinking time will increase.
Vehicle Braking Distance
When you apply the brakes your vehicle will slow and eventually stop.
The distance travelled while braking is governed by
the laws of physics and how well your tyres grip the road,
assuming of course that you have good brakes and a strong right leg.
This chart also assumes a dry road -
braking distances on a wet road can be more than double these.
Obviously, the faster you travel the greater the distance you will travel in a given time. This graph shows how far, in metres, you will travel in a few seconds at various speeds, in miles per hour.
From the first chart, you will note that the shortest stopping distance at 50 miles per hour is 53m. From the graph on the right you will see that when travelling at a constant 50 miles per hour you will cover this 53m in just under two and a half seconds (2.4s).
From now on all graphs will use this colour scheme.
For example, a light blue line is always used
for an initial speed of 50 miles per hour (50 mph).
This graph shows how far you will travel while stopping. For the first 0.7 seconds this graph is the same as the pervious one, because this is your thinking time when you are still travelling at your initial speed.
After 0.7 seconds the lines bend as your speed decreases as you brake. Eventually, when you have stopped the lines become horizontal.
At 70 mph it will take you about five and a half seconds to stop.
Unlike the curved lines for distance, the graph of speed vs. elapsed time has only straight lines. Note the bends at 0.7 seconds.
If you were travelling at 30 mph then
three seconds after you saw a hazard
you would have just stopped.
If you were travelling at 50 mph then
three seconds after you saw a hazard
you would still be going at 16 mph.
This is very informative graph. It is the same as the pervious graph except that the horizontal time axis has been replaced by distance. It shows your speed against the distance you have travelled. As in the graph above, there are kinks at 0.7 seconds.
Suppose the hazard was 12.5 metres away when you first saw it.
If you were travelling at 20 mph then
you will stop safely in less than 12.5m.
If you were travelling at 40 mph then
your foot has not even have touched the brake before you have travelled that 12.5m!
At 20 mph there is no collision,
but at 40 mph you hit that child at 40 mph!
This graph is similar to the previous one, except for the horizontal axis. The horizontal axis still measures distance, but now the distance is expressed as the time it would take you to cover that distance at your initial speed.
Do you use the "two-second rule"? The rule is always to keep a gap of at least two seconds between you and the vehicle in front. Using this graph you can see that if you left a two second gap and the vehicle in front stopped almost instantly, as it might do if someone pulled out of a side road, then at 30 mph you would stop safely, at 40 mph you would hit the wreck at 9 mph, but at 70 mph you would hit the wreck at 47 mph.
Two seconds is about how it takes to say "only a fool breaks the two-second rule". When the vehicle in front passes a landmark, for example a bridge, say to yourself "only a fool breaks the two-second rule". If you reach the landmark before you have finished then you're too close.
The "two-second rule" is useful, as many drivers find it easier to judge time than distance. Using this rule, as your speed increases so the gap distance will increase. However, at high speeds you should increase the gap to three seconds, and to six or more if the roads are wet.
There are two ways to increase the gap to meet the two-second rule,
a) slow for a while then speed up to your original speed, or
b) slow (because you require a shorter distance at a lower speed).
The other alternative is not so good
c) slow forever (because you're dead).
This chart is the same as the first one, with two additional straight lines for 2 seconds (green) and 3 seconds (red). As you can see, a two second gap is by no means excessive compared to your stopping distance.
Oh yes, never leave a gap of less than your thinking distance. Your thinking distance, or 0.7 seconds, is the absolute minimum you require to have even a chance of avoiding an accident.
Happy, safe, motoring.
The data for this analysis was taken from pages 28 and 29 of the 1999 "Highway Code". This has the classic graph of braking distances at various speeds, showing the "typical stopping distances".
| Speed | Thinking Distance | Braking Distance | Stopping Distance |
|---|---|---|---|
| 20 | 6 | 6 | 12 |
| 30 | 9 | 14 | 23 |
| 40 | 12 | 24 | 36 |
| 50 | 15 | 38 | 53 |
| 60 | 18 | 55 | 73 |
| 70 | 21 | 75 | 96 |
1 mph = 0.447 m/s (1 mile = 1609m, 1 hour = 3600 seconds)
Some definitions
Physics
The equations of motion determine how a vehicle will behave under constant acceleration.
| a | = constant (= F/m) | ||||
| Integrating this with respect to time we get | |||||
| v | = at + v0 | [1] | |||
| Integrating again we get | |||||
| x | = at2/2 + v0t + x0 | [2] | |||
There are three distinct periods of time,
the thinking time, when the velocity is constant, a = 0.
the braking time, when the velocity is decreasing, a = a negative constant.
the stopped time, when the velocity is zero, v = 0, a = 0.
Thinking
Substituting a=0 into [1] and [2], during thinking time we have
| v | = v0 | [3] | |||
| x | = v0t | [4] |
| t' | = t - 0.67 | [5] | |||
| x'0 | = 0.67v0 | [6] |
Braking
| v' | = at' + v'0 | (from [1] ) | |||
| v | = a(t - 0.67) + v0 | [7] | (from [5] ) | ||
| x' | = at'2/2 + v0t' + x'0 | (from [2] ) | |||
| x | = a(t-0.67)2/2 + v0(t-0.67) + 0.67v0 | (from [5] and [6] ) | |||
| = a(t-0.67)2/2 + v0t | [8] |
Stopped
By definition we stop when v = 0. This is when t = T
| 0 | = a(T - 0.67) + v0 | (from [7] ) | |||
| T | = 0.67 - v0 / a | [9] | |||
| xT | = v0T + a(T - 0.67)2/2 | (from [8] ) | |||
| = 0.67v0 - v02/2a | [10] | (from [9] ) |
| Thinking | t <= 0.67 | v = v0 | [3] | x = v0t | [4] | |||
| Braking | t >= 0.67 t <= 0.67 - v0 / a |
v = a(t - 0.67) + v0 | [7] | x = a(t-0.67)2/2 + v0t | [8] | |||
| Stopped | t >= 0.67 - v0 / a | v = 0 | x = 0.67v0 - v02/2a | [10] | ||||
| where a = -6.56 m/s/s | ||||||||