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

DESCRIPTION	EQUATION
Thinking distance	$thinking \ distance = reaction \ time \times speed$
Stopping distance	$stopping \ distance = thinking \ distance + braking \ distance$

Thinking, braking and stopping distances

$Speed \ (Mph)$	$20$	$30$	$40$	$50$
$Speed \ (ms^{-1})$	$8.9$	$13.4$	$17.8$	$22.2$
$Thinking \ distance \ (m)$	$6$	$9$	$12$	$15$
$Braking \ distance \ (m)$	$6$	$14$	$24$	$38$
$Stopping \ distance \ (m)$	$12$	$23$	$36$	$53$

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:

FACTOR	EXPLANATION
Driver's speed	The faster the driver is going, the greater the distance the vehicle will cover for the same reaction time, as can be seen in the table above.
Driver's reaction time	The longer the driver's reaction time, the longer their thinking distance. This can be affected by tiredness and being under the influence or drugs.

Physics; Motion; KS5 Year 12; Speed and stopping distances

Physics; Motion; KS5 Year 12; Speed and stopping distances

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.

FACTOR	EXPLANATION
Vehicle speed	The faster the driver is going, the longer it takes to stop, as seen in the table at the beginning of the summary
Weather or road surface	If there is less grip between the vehicle's tyres and the road, it can cause the vehicle to skid, which increases the braking distance of the vehicle. Water, ice, oil or leaves on the road surface all reduce the available grip of the vehicle's tyres, making it more likely to skid.
Tyre condition	Tyres that have no tread left cannot remove water in wet conditions effectively, which can cause the vehicle to skid.
Brake condition	Worn brakes will not be able to apply as much braking force, meaning the vehicle will take longer to stop.
Mass of vehicle	A more massive vehicle will have a larger braking distance as it will have more kinetic energy to dissipate in the brakes.

Physics; Motion; KS5 Year 12; Speed and stopping distances

Physics; Motion; KS5 Year 12; Speed and stopping distances

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.

Physics; Motion; KS5 Year 12; Speed and stopping distances

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.

Example:

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 $\underline{0.68 \ s}$ .