Stopping Distance Simulator

What is the vehicle stopping distance formula?

Total vehicle stopping distance is the sum of thinking distance and braking distance. On a level road: Stopping Distance = (v · t_r) + [v² / (2 · μ · g)], where v is initial speed in m/s, t_r is driver reaction time in seconds, μ is tyre-road friction coefficient, and g = 9.81 m/s² is the acceleration due to gravity.

A stop has two parts. First the driver has to see the hazard and move a foot to the brake, and for that reaction time the car carries on at full speed: the thinking distance, v t_r. Then the brakes slow it at a steady deceleration a until it is at rest, and the equation of motion v² = u² + 2as, with a final speed of zero, gives the braking distance, v²/2a. That is the equation the kinematics calculator solves. On a level road the tyres can give a deceleration of at most μg, where μ is the coefficient of friction between tyre and road, so the stopping distance is

• S = v t_r + v²/(2μg)

The simulator opens on the Highway Code’s typical car, at 50 km/h with a reaction time of 0.67 s and μ = 0.667. Worked through:

1. The speed in metres per second. 50 km/h ÷ 3.6 = 13.89 m/s.
2. The thinking distance. 13.89 m/s × 0.67 s = 9.31 m.
3. The deceleration. μg = 0.667 × 9.81 m/s² = 6.54 m/s², about two thirds of g.
4. The braking distance. (13.89 m/s)² ÷ (2 × 6.54 m/s²) = 14.75 m.
5. The stopping distance. 24.05 m, about six car lengths of 4 m. That is the sum of the unrounded parts, 9.306 m and 14.745 m; the rounded figures above add to 24.06 m.

The time to stop is the reaction time plus the braking time, v/a: 0.67 s + 2.12 s = 2.79 s. The simulator’s two cars play that out on the road, the reader’s in the near lane and a comparison car, the same driver 10 km/h faster, in the far lane. The road behind each is painted amber while its driver reacts and red, hatched, while it brakes.

Why speed counts twice

Speed appears once in the thinking distance and twice in the braking distance. Thinking distance grows in proportion to speed, but braking distance grows with its square, because the brakes have to remove the car’s kinetic energy, ½mv², with a friction force that does not grow with speed. That square is the one the kinetic energy calculator shows: twice the speed is four times the energy. So doubling the speed doubles the thinking distance and quadruples the braking distance. At 100 km/h the Highway Code’s car needs 18.61 m of thinking and 58.98 m of braking, 77.59 m in all: 3.2 times as far as at 50 km/h, not twice.

The upper plot draws it. Thinking distance is a straight line and braking distance a curve that bends upwards, and they cross where v t_r = v²/2a, at v = 2at_r. With the Highway Code’s numbers that is 8.76 m/s, 31.6 km/h: above it, most of any stop is braking.

The crossing makes the same point. At the opening settings the 50 km/h car stops 0.95 m short of the zebra crossing 25 m ahead. The same driver at 60 km/h, a fifth faster, covers 1.86 m more before the brakes come on and needs (60/50)² = 1.44 times the braking distance, 21.23 m against 14.75 m. By the time its driver reacts it has only 13.83 m left in which to brake, and it reaches the crossing still doing 35.4 km/h, with more than a third of the kinetic energy it started with. The momentum calculator gives its momentum at that speed for any mass.

The Highway Code table, rebuilt from two numbers

Rule 126 of the Highway Code gives typical stopping distances from 20 to 70 mph. Its thinking distances are exactly 0.3 m for every mph, which is a reaction time of 0.671 s, and its braking distances are fitted best by a single deceleration of 6.54 m/s². Set UK units, the Highway Code car and 0.67 s, and every row comes back to the metre:

The Code counts a car as 4 m and calls these distances typical. The same rule says the gap should be at least doubled on wet roads and up to ten times greater on icy roads, and Rule 227 gives the reason for the first: tyres have less grip on a wet road.

Wet, icy and sloping roads

Grip is the coefficient of friction μ, and the simulator’s surfaces take the middle of each range of road adhesion that J. Y. Wong tabulates in Theory of Ground Vehicles. Each surface has two figures: the peak a rolling tyre reaches just before it slips, which ABS holds the tyre at, and the lower sliding value once the wheels lock. At 50 km/h:

From 11.6 m on dry asphalt, braking with ABS grows to 16.4 m on a wet road and 98.4 m on ice. Locking the wheels costs grip on every one of these surfaces, and a car whose front wheels lock cannot be steered, which is why ABS exists. On loose gravel and fresh snow the order can reverse, because a locked tyre ploughs up a wedge of material ahead of it, which no single coefficient describes.

A slope changes the deceleration to g(μ cos θ + sin θ), where tan θ is the grade over 100, uphill positive. Uphill, part of the car’s weight pulls it back and helps the brakes; downhill, it pushes the car on. This is the resolving of forces the inclined plane simulator draws as a free-body diagram. At 50 km/h with the Highway Code’s car, braking takes 14.75 m on the level, 17.43 m down a 10 percent grade and 12.89 m up one.

On ice the effect can be total. With ABS on ice μ is 0.10, and down a 10 percent grade tan θ is 0.10 as well, so μ cos θ equals sin θ and the deceleration is zero: the car slides on at the speed it had, however long the brakes are held. With the wheels locked, μ is 0.07 and the car speeds up. Choose Downhill on ice to watch the first.

Reaction time is not one number

The thinking distance is the reaction time times the speed, so at 50 km/h every tenth of a second costs 1.39 m. A ruler-drop test usually measures about 0.2 s: catch a falling ruler after 21 cm and it fell for 0.21 s. That is a reaction to a signal you were waiting for. Braking for a hazard you did not expect takes longer. In Marc Green’s review of the evidence (Transportation Human Factors, 2000), drivers took about 0.7 s to brake when they expected the signal, about 1.25 s for an unexpected but common one, such as brake lights ahead, and about 1.5 s when surprised. At 50 km/h, 1.5 s is 20.83 m of thinking distance, more than twice the Highway Code’s 9.31 m.

Measure your own with the reaction time ruler-drop experiment, which can send its result here as the reaction time.

Trucks, motorcycles and mass

Mass cancels from the stopping distance. A heavier vehicle presses its tyres onto the road harder, so the most friction they can give, μmg, grows in proportion to its mass, and so does the force it needs to slow down. Switch the vehicle from the car to the motorcycle and every distance stays the same. Only the energy the brakes turn to heat changes, from 144.7 kJ for a nominal 1,500 kg car to 24.1 kJ for a nominal 250 kg motorcycle and rider. The motorcycle has no braking figure of its own here: its deceleration is whatever the grip allows. On the road, Rule 126 of the Highway Code warns that motorcycles, like large vehicles, need a greater distance to stop.

Trucks stop further for a different reason: their brakes, not their tyres, set the limit. US federal rules (FMVSS 121) require most loaded truck tractors to stop from 60 mph within 250 ft, measured from the moment the brake pedal starts to move, an average deceleration of 4.72 m/s², 0.48 g. A few of the heaviest are allowed 310 ft. The rule for cars (FMVSS 135) allows 0.10V + 0.0060V² metres from V km/h, and its V² term is a deceleration of 6.43 m/s², close to the Highway Code’s 6.54. The simulator’s loaded truck brakes no harder than 4.72 m/s² on any surface that grips better than that, so from 60 mph, with the Highway Code’s 0.67 s, it needs 94.17 m, 309 ft, where the Highway Code’s car needs 72.97 m.

Worked exam question: 126 km/h stopped within 200 m

An exercise in NCERT’s Class 11 physics textbook brings a car at 126 km/h to a stop within 200 m and asks for its retardation, taken as uniform, and the time it takes to stop. The question counts only the braking.

1. The speed. 126 km/h ÷ 3.6 = 35 m/s.
2. The retardation. From v² = u² + 2as with v = 0, a = u²/2s = 35² ÷ (2 × 200) = 3.06 m/s².
3. The time. From v = u + at, t = u/a = 35 ÷ 3.0625 = 11.4 s.

Choose Exam: 126 km/h in 200 m to set it up: the braking is worked out from a braking distance of 200 m, the reaction time is 0 s, and the readouts give 3.06 m/s² and 11.4 s. Then give the driver the Highway Code’s 0.67 s to see what a real reaction adds: 23.45 m before the brakes come on.

Reading the two plots

The upper plot is distance against speed for this driver, road and vehicle, from a standstill to the top of the speed slider, with the two cars marked as dots and, when the Highway Code car and 0.67 s are set, the Code’s own totals. The lower plot is speed against time for this stop: flat for the reaction time, then a straight fall to rest, because the deceleration is constant. The area under each line is that car’s stopping distance, the reading of a speed-time graph the motion graphs simulator is built around.

What this model leaves out

• Brake build-up. Real brakes take a fraction of a second to reach full force. The Highway Code’s figures fold that into one average deceleration, and so does this model, which brakes fully the moment the reaction ends.
• Grip that changes with speed. One coefficient serves at every speed. On standing water a tyre can ride up on the water and lose contact, which is aquaplaning, and then no coefficient here describes the stop.
• Brake fade. Long or repeated stops heat the brakes until they grip less, which matters most for loaded trucks on long descents.
• Drag, rolling resistance and engine braking. Each would shorten a stop a little, most at motorway speeds.
• Steering. The vehicle brakes in a straight line. Steering while braking uses grip that the model gives entirely to stopping, and a vehicle that skids sideways is outside it.
• Loose surfaces. On loose gravel and fresh snow, ABS can lengthen a stop rather than shorten it.

Common mistakes

• Leaving the speed in km/h. v²/2a needs metres per second: 50² ÷ (2 × 6.54) gives 191, not 14.75. Divide km/h by 3.6 first.
• Forgetting the thinking distance. At 50 km/h it is 9.31 m of the 24.05 m, more than a third of the stop.
• Expecting twice the speed to need twice the distance. From 50 to 100 km/h the whole stop grows 3.2 times, and the braking part 4 times.
• Using a μ above 1. Ordinary tyres on dry asphalt reach about 0.8 to 0.9. Figures above 1 belong to soft performance and racing tyres.
• Reading a stopping distance as a safe gap. These are the distances of one model vehicle. The Highway Code calls its own figures typical and tells drivers to leave more room in the wet and on ice.

Constants used: Standard gravity (g₀).