Library / Automotive and vehicle dynamics / VEH-12
Automotive and vehicle dynamics
Total stopping distance with reaction time
Distance to stop from speed U₀: the distance covered during the reaction or response time at constant speed, plus the braking distance at constant deceleration.
- Ref
- VEH-12
- Rev
- v1
- Field
- Automotive and vehicle dynamics
- Result
- sstop [m]
U0 is the speed when the driver reacts (entered in km/h and converted to m/s), tr the reaction or system response time, and ab the constant braking deceleration entered as a multiple of g. The vehicle keeps its speed during the reaction time and then decelerates uniformly to rest. The deceleration depends on the surface and tyres: about 0.8 g on dry tarmac, 0.5 g on wet and 0.15 g on ice.
Distance to stop from speed U₀: the distance covered during the reaction or response time at constant speed, plus the braking distance at constant deceleration.
Calculator
- Reaction distance, U₀ tr27.78 m
- Braking distance, U₀² / 2ab49.16 m
- Time to stop4.539 s
- Speed in m/s27.78 m/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- From 100 km/h at 0.8 g with a 1 s reaction time the car travels 27.78 m before braking and 49.16 m while braking, 76.94 m in all. The reaction share is 36.1 % of the total.
- Doubling the speed to 200 km/h gives 252.2 m, more than three times as far, because the braking part grows with the square of the speed. On a wet road (0.5 g) the same stop from 100 km/h needs 106.4 m.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| U0 Initial speed | 10 to 250 km/h | 3.269 | 376.7 | 373.4 | +10% | +17 % |
| ab Braking deceleration | 0.1 to 1.5 g | 54 | 421.1 | 367.1 | +10% | −5.81 % |
| tr Reaction or response time | 0 to 3 s | 49.16 | 132.5 | 83.33 | +10% | +3.61 % |
Sweep of Initial speed
| Initial speed (km/h) | Total stopping distance (m) |
|---|---|
| 10 | 3.269 |
| 40 | 18.98 |
| 70 | 43.53 |
| 100 | 76.94 |
| 130 | 119.2 |
| 160 | 170.3 |
| 190 | 230.2 |
| 220 | 299 |
| 250 | 376.7 |
Sweep one variable
Variables and how each one contributes
Total stopping distance
What the equation gives you. Everything else feeds it.
Initial speed
The speed at the moment braking is needed, in km/h. The braking distance grows with the square of the speed and the reaction distance linearly, so the total grows faster than proportionally. Raising it by 10 % (to 110) moves the result by +17 %, from 76.94 to 90.04 m.
Reaction or response time
The time between the need to brake and the brakes taking effect, in s: about 1 s for an alert driver, longer if distracted, and shorter for an automatic system. Adds a distance of U₀ tr: at 100 km/h each extra second adds 27.8 m. Raising it by 10 % (to 1.1) moves the result by +3.61 %, from 76.94 to 79.71 m.
Braking deceleration
The constant braking deceleration, as a multiple of g (1 g = 9.81 m/s²). It cannot exceed the tyre–road friction coefficient. Inversely proportional in the braking distance: harder braking shortens it, and good tyres matter on wet roads. Raising it by 10 % (to 0.88) moves the result by −5.81 %, from 76.94 to 72.47 m.
Worked example with the values above
- Speed: U0 = 100 / 3.6 = 27.78 m/s. Deceleration: ab = 0.8 × 9.81 = 7.848 m/s².
- Reaction distance: U0 tr = 27.78 × 1 = 27.78 m. Braking distance: U0² / (2ab) = 771.6 / 15.7 = 49.16 m.
- sstop = 27.78 + 49.16 = 76.94 m.
Assumptions and limits
- Constant deceleration from the moment the brakes take effect, and a constant speed during the reaction time.
- A level road with no wind, and the same deceleration for the whole stop, ignoring brake build-up, ABS cycling and speed-dependent drag.
Common mistakes
- Entering the speed in m/s while the box expects km/h.
- Using the braking distance alone for safety distances. The reaction distance is often the larger part at low deceleration.
Statistics and uncertainty
The two parts have different uncertainties. The reaction time varies by at least ±0.3 s between drivers, adding ±8 m at 100 km/h, and the deceleration depends on the surface, so a 20 % error in the friction coefficient changes the braking distance by about 20 %. Measured stopping distances from a controlled test are good to a few per cent; the road-use value has a far bigger spread.
Where it is used
- Road safety, following distances and design speeds.
- Brake performance requirements and the stopping distance of military and emergency vehicles.
Maths you need
- Constant acceleration: v² = u² + 2 a s
- Units: km/h to m/s