Same Brakes, Twice the Speed
Science, Worked Out — from Arxo. Episode 01.
A 1,500 kg car brakes hard from 14 m/s, about 50 km/h, and stops in 14 metres. Same car, same brakes, twice the speed: 28 m/s, about 100 km/h. How far does it travel before it stops?
Make a guess before you read on.
If you said 28 metres, you doubled the distance along with the speed. The worked answer is 56 metres. Twice the speed takes twice the time to stop and four times the distance.
Two quantities, two answers
The same motion has both a momentum and a kinetic energy. Momentum is mass times velocity: p = mv. Kinetic energy is half the mass times the speed squared: K = ½mv². Double the speed, and they part ways.
| Car of 1,500 kg | at 14 m/s | at 28 m/s |
|---|---|---|
| Momentum, p = mv | 21,000 kg·m/s | 42,000 kg·m/s |
| Kinetic energy, K = ½mv² | 147,000 J | 588,000 J |
Momentum doubles. Energy quadruples, because the speed enters it twice.
Momentum tells us about the stopping time. Kinetic energy tells us about the braking distance.
Momentum sets the time, energy sets the distance
Assume a constant net braking force of 10,500 newtons, opposite to the motion. For a 1,500 kg car, that force gives an acceleration of −7 m/s²: the car loses 7 m/s of speed every second.
A constant force removes momentum at a steady rate: every second takes away the same amount. Twice the momentum under the same force takes twice the time. To check this, we picked a moment for each start by dividing the starting speed by 7 m/s², which gives 2 s for 14 m/s and 4 s for 28 m/s. Then we asked Arxo for the car's speed at that moment. Both times the answer is exactly 0 m/s.
A constant force also removes energy at a steady rate per metre: every metre takes away the same amount. Four times the energy under the same force takes four times the distance. Asked how far the car travels by those moments, Arxo answers 14 m for the slower start and 56 m for the faster one.
The run checks itself in one more place. Using the 14 metres from the motion calculation, Arxo computes the work done by the braking force: −147,000 J. That is the car's whole kinetic energy at 14 m/s, with a minus sign because the brakes take it away. Over 56 metres the work is −588,000 J, again the negative of the starting energy. Work and energy come from different rules applied to different facts, and they match exactly.
Now change the mass
Keep the speed at 14 m/s and the braking force at 10,500 N, and double the mass to 3,000 kg. The same brakes now have twice as much to stop.
| Same brakes, 14 m/s | 1,500 kg | 3,000 kg |
|---|---|---|
| Acceleration | −7 m/s² | −3.5 m/s² |
| Momentum | 21,000 kg·m/s | 42,000 kg·m/s |
| Kinetic energy | 147,000 J | 294,000 J |
| Moment we asked about (our choice) | 2 s | 4 s |
| Speed at that moment | 0 m/s | 0 m/s |
| Braking distance | 14 m | 28 m |
Doubling the mass doubles both the momentum and the energy, so the time and the distance double together. Doubling the speed doubles the momentum but quadruples the energy. Of the two inputs, only the speed appears squared.
The comparison holds as long as the braking force stays the same. There is another common idealisation: a grip-limited stop with a constant friction coefficient. In that model the available braking force grows in proportion to the mass, the acceleration stays the same, and the mass drops out of the braking distance. That is a different model of the stop, and it answers a different question.
The same square in the Highway Code
The UK Highway Code publishes a table of typical stopping distances, which it describes as "a general guide". Its braking distances at 20 and 40 mph show the same pattern: 6 m and 24 m, twice the speed and four times the distance.
The table adds a second part, the thinking distance: how far the car travels before the driver starts to brake. That part is 6 m at 20 mph and 12 m at 40 mph. With the same reaction time, doubling the speed doubles the thinking distance. Together, the two parts grow from 12 m to 36 m.
Source: The Highway Code, rule 126, and its typical stopping distances table (PDF), accessed 11 October 2026. Contains public sector information licensed under the Open Government Licence v3.0.
Try it yourself
Pick a mass and a speed, guess the braking distance, then reveal the worked answer. The page shows the rule behind each value and the result hash that identifies its run.
Try it yourself: same brakes, five speeds, two masses
What was computed, what was chosen, what was assumed
- Computed by Arxo. The accelerations, momenta and kinetic energies, the speed at each moment we asked about, the braking distances and the work of the brakes. Arxo applied six rules to stated inputs: p = mv, K = ½mv², a = F/m, v = v₀ + at, x = v₀t + ½at² and W = Fd. Each answer comes with its derivation and a result hash, which changes when any input or rule changes. When all the questions were run again on 11 October 2026, every hash came out identical.
- Chosen by us. The masses, the starting speeds, the braking force and the moments 2 s and 4 s. The km/h figures are our rounding.
- Assumed. One constant net braking force, measured in the frame of the road, on a straight level road at everyday speeds. The model covers the car up to the moment it first stops, and it leaves out reaction time, air resistance, the rotation of the wheels, tyre grip and brake fade.
- Published. The Highway Code figures are quoted from its official table, as a comparison with published guidance.
The facts, the questions, all 52 answers and the full derivation of each one are available as files.
Worked out with Arxo. Models formalize; Arxo executes and proves.