When a car brakes to rest, the brakes exert a force over the braking distance. Work done equals force multiplied by distance, W = F × d. The car stops when the brakes have taken away all of its kinetic energy as thermal energy, so the work done by the brakes equals the initial kinetic energy.
The kinetic energy of a moving object is ½ × m × v2, with mass m in kg, velocity v in m/s and energy in joules. Putting the two ideas together gives F × d = ½ × m × v2, so the braking distance is d = (½ × m × v2) ÷ F.
Because the velocity is squared, doubling the velocity makes the kinetic energy four times bigger. For the same braking force the braking distance is therefore four times bigger too. Tripling the velocity makes the braking distance nine times bigger.
Worked example: a car of mass 1000 kg travels at 20 m/s. Its kinetic energy is ½ × 1000 × 202 = 200 000 J. A constant braking force of 5000 N stops it in 200 000 ÷ 5000 = 40 m. At 30 m/s the kinetic energy is 450 000 J, so the braking distance is 90 m. The speed rose by a factor of 1.5, and the distance rose by 2.25, which is 1.5 squared. A larger mass needs a longer distance for the same force, and a weaker braking force, for example on a wet road, also needs a longer distance.