Speed and stopping distances

​​In a nutshell

The stopping distance of a car is the sum of the thinking distance and braking distance. The thinking distance is distance travelled between the moment a hazard is seen, and the moment brakes are applied. The braking distance is the distance travelled between the brakes being applied and the vehicle coming to a complete stop.

​

Equations

Thinking, braking and stopping distances

Note: you do not need to remember these values.

Thinking distance

The thinking distance is the distance travelled during a driver's reaction time. The reaction time is defined as the time between seeing a hazard and applying the brakes.

The thinking distance is proportional to the initial speed the car is travelling at, the faster the car is travelling, the longer the stopping distance, as can be seen in the table above. It can be calculated using the equation:

$$thinking \ distance = reaction \ time \times speed$$

The thinking distance is proportional to the speed the driver is travelling at, as reaction time is constant without external factors, which are shown below:

Note: the average human reaction time is $$0.2 \ s$$.

​

Braking distance

The braking distance is the distance travelled between the brakes being applied and the vehicle coming to a complete stop.

To stop the vehicle, the brakes must do work to transfer all the energy from the vehicles kinetic store:

​$$W = E_k\newline \\[0.1in] Fd = \frac{1}{2} mu^2 \newline \\[0.1in] d \propto u^2$$​​

The braking distance of the car is proportional to the initial speed of the car squared, $$u^2$$, as the maximum braking force and mass of the vehicle will be constant.

Stopping distance

The stopping distance of a car is defined as the total distance from when the driver sees the hazard to when the vehicle comes to a complete stop. The stopping distance consists of the thinking distance and braking distance, as shown in the word equation:

$$stopping \ distance = thinking \ distance + braking \ distance$$

In general, the heavier the vehicle, the longer the stopping distance will be.

Stopping distance graphs

The following graph is a stopping distance graph, which is a form of velocity-time graph. The  total stopping distance travelled can be obtained by calculating the area under the graph. 

When the graph is flat, this represents the thinking distance as the driver has not applied their brakes, and therefore not slowed down. When there is a slope the driver has applied the breaks as they are now slowing down.

In the UK Highway Code, the thinking distance at $$50 \ mph \ (22.2 \ ms^{−1})$$​ is shown as $$15 \ m$$​. Calculate the corresponding reaction time.

​

State variables:

$$speed = 22.2 \ ms^{-1}, \ thinking \ distance = 15m$$

State equation:

$$thinking \ distance = reaction \ time \times speed$$

Sub in and solve:

$$reaction \ time = \dfrac{thinking \ distance}{speed}\newline \\[0.1in] reaction \ time = \dfrac{15}{22.2}\newline \\[0.1in] reaction \ time = 0.68 \ s$$​​

The reaction time is $$\underline{0.68 \ s}$$.

​

​​