Stopping distance: the braking distance and thinking distance formulas

The braking distance formula is d = v² ÷ 2a, where v is the speed when the brakes are applied and a is the deceleration, from the equation v² − u² = 2as. Put another way, the brakes must remove all of the car's kinetic energy, so Fd = ½mv². Add the thinking distance, speed × reaction time, and you get the stopping distance. A car at 20 m/s with a 0.7 s reaction time and a deceleration of 6.5 m/s² travels 14 m before braking and about 31 m while braking: 45 m in all.

Stopping distance = thinking distance + braking distance

AQA's GCSE Physics specification defines the stopping distance as the sum of the distance the vehicle travels during the driver's reaction time (the thinking distance) and the distance it travels under the braking force (the braking distance). For a given braking force, the greater the speed, the greater the stopping distance.

The thinking distance formula

During the reaction time the car keeps going at its original speed, so:

AQA says reaction times vary from person to person, with typical values from 0.2 s to 0.9 s, and can be affected by tiredness, drugs and alcohol. Distractions can also affect a driver's ability to react.

The braking distance formula

Once the brakes are on, the car decelerates from speed v to rest. With u = v, final velocity 0 and deceleration a, the equation v² − u² = 2as from AQA's equations sheet gives:

The energy version

AQA's specification says that when the brakes are applied, work done by the friction force between the brakes and the wheel reduces the kinetic energy of the vehicle, and the temperature of the brakes increases. Setting work done equal to the kinetic energy removed:

Both versions say the same thing, since F = ma. The key point is the v²: double the speed and the braking distance goes up four times, while the thinking distance only doubles.

Here x is the speed in m/s and y is the stopping distance in m, for a reaction time of 0.7 s and a deceleration of 6.5 m/s². The straight part of the rise is the thinking distance; the curve comes from the braking distance. Three times the speed, 30 m/s instead of 10 m/s, gives a stopping distance about six times longer. AQA expects GCSE Physics students to interpret graphs like this for a range of vehicles.

Stopping distance worked examples

Example 1: thinking distance

A driver travelling at 20 m/s has a reaction time of 0.7 s. Find the thinking distance.

Example 2: braking and stopping distance

The same car decelerates at 6.5 m/s² once the brakes are applied. Find the braking distance and the total stopping distance.

Stopping distance = 14 + 30.8 = 44.8 m.

Example 3: the braking force

A 1100 kg car travelling at 24 m/s stops in a braking distance of 48 m. Find the average braking force.

The car's kinetic energy is ½ × 1100 × 24² = 316 800 J. The brakes do this much work over 48 m:

Example 4: what doubling the speed does

A driver with a reaction time of 0.6 s brakes at 7.5 m/s². Compare the stopping distances at 15 m/s and 30 m/s.

At twice the speed, the thinking part is 2 times as long and the braking part 4 times, so the total more than triples.

Example 5: the effect of a slower reaction

At 13 m/s, a driver's reaction time rises from 0.5 s to 0.9 s after drinking alcohol. How much further do they travel before braking?

Thinking distance goes from 13 × 0.5 = 6.5 m to 13 × 0.9 = 11.7 m: 5.2 m further, before the brakes are even touched.

Now try one in the course. Choose the equation of motion that fits, then press Mark to check it there.

The SUVAT Calculator solves v² = u² + 2as and the other equations of motion from any three values, so you can check braking distances and decelerations. Our guide to the SUVAT equations covers all five.

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Factors that affect stopping distance

AQA lists tiredness, drugs, alcohol and distractions as affecting reaction time, and adverse road and weather conditions (wet or icy roads) and poor condition of the vehicle (brakes or tyres) as affecting braking distance. Speed affects both. A heavier vehicle, with the same braking force, also takes longer to stop, because Fd = ½mv² gives a bigger d for a bigger m.

Why large decelerations are dangerous

AQA's specification says that the greater the braking force, the greater the deceleration, and that large decelerations may lead to brakes overheating and/or loss of control. Braking hard transfers a lot of kinetic energy to the brakes very quickly.

Common mistakes

• Forgetting the thinking distance. Stopping distance includes both parts.
• Using the wrong speed in the thinking distance. It is the speed before braking starts.
• Saying braking distance doubles when speed doubles. It quadruples, because it depends on v².
• Saying alcohol affects braking distance. It affects reaction time, so thinking distance.
• Saying wet roads affect thinking distance. They affect braking distance.
• Speed in km/h or mph. Convert to m/s first: divide km/h by 3.6.

Practise forces in a short timed paper

GCSE Physics Higher and GCSE Physics Foundation each give you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so stopping distances turn up regularly but not in every paper.

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