Speed and thinking distance

1. Thinking distance is the speed multiplied by the reaction time, and the reaction time does not depend on how fast the vehicle is going.
2. So for the same driver, thinking distance is directly proportional to speed, and doubling the speed doubles the thinking distance.
3. A graph of thinking distance against speed is a straight line through the origin.
4. With a typical reaction time of about $0.7s$, a car at $13m/s$ has a thinking distance of about $9m$, and a car at $30m/s$ about $21m$.

Speed and braking distance

1. To stop, the brakes must transfer all of the vehicle's kinetic energy, and kinetic energy $=\frac{1}{2}m{v}^2$ depends on the square of the speed.
2. Doubling the speed gives the vehicle four times as much kinetic energy, and tripling the speed gives nine times as much.
3. The work done by the brakes is $braking\; force\times braking\; distance$, and the greatest braking force the tyres can provide is about the same at every speed.
4. So the braking distance is proportional to the square of the speed, and doubling the speed makes the braking distance about four times longer.
5. A graph of braking distance against speed is a curve that gets steeper as the speed increases.
6. The same result comes from $v^2-u^2=2as$, which for a vehicle stopping from speed $u$ with a deceleration of size $a$ gives a braking distance of $\frac{u^2}{2a}$.
7. A typical deceleration in an emergency stop on a dry road is about $6$ to $7{m/s}^2$, so this equation gives a quick estimate of braking distance at any speed.

Total stopping distance

1. Stopping distance is the sum of a part that rises in proportion to speed and a part that rises with the square of the speed.
2. At low speeds the thinking distance is a large share of the total, but at high speeds the braking distance is by far the larger part.
3. So the stopping distance rises more steeply than the speed, and doubling the speed roughly triples the stopping distance at typical road speeds.
4. The typical stopping distances for a car on a dry road, as used in the UK Highway Code, are listed below.
   1. 20 mph ($9m/s$): thinking distance $6m$, braking distance $6m$, stopping distance $12m$.
   2. 30 mph ($13m/s$): thinking distance $9m$, braking distance $14m$, stopping distance $23m$.
   3. 40 mph ($18m/s$): thinking distance $12m$, braking distance $24m$, stopping distance $36m$.
   4. 50 mph ($22m/s$): thinking distance $15m$, braking distance $38m$, stopping distance $53m$.
   5. 60 mph ($27m/s$): thinking distance $18m$, braking distance $55m$, stopping distance $73m$.
   6. 70 mph ($31m/s$): thinking distance $21m$, braking distance $75m$, stopping distance $96m$.
5. A stopping distance of $96m$ at $70$ mph is about the length of a football pitch.
6. These figures assume an alert driver and a well-maintained car on a dry road, so real stopping distances in the rain or with a tired driver are longer.
7. Between two listed speeds, a sensible estimate lies between the two listed stopping distances, a little below the halfway value because the curve gets steeper.