Speed from tire marks and drag factor

Three days after the collision, on a dry Tuesday morning with Kingsway closed for an hour, Kowalczyk drove a borrowed pickup of the same make, model and year as Halloran's along the westbound curb lane and braked as hard as he could at a cone. Constable Grace Teague, the unit's other reconstructionist, measured the stopping distance and checked the speed against a calibrated test unit. They did it again, and then a third time to check the first two. From those few metres of road, they would calculate the single most important number in the case: how fast Halloran's truck was travelling when its driver first hit the brakes.

This lesson explains the physics behind speed estimates from tire marks. It is the most mathematical lesson in the course, but every step uses one idea that most students already know: a moving vehicle that brakes hard slows down at a rate set by the grip between its tires and the road. If investigators know that rate and how far the vehicle slid, they can work out how fast it was going when the braking began.

Drag factor

The drag factor, written f, is the deceleration of a braking vehicle expressed as a fraction of the acceleration due to gravity. A drag factor of 0.72 means the vehicle slows at 0.72 times g, where g is about 9.81 metres per second squared. The drag factor depends on the road surface, its condition, the tires and the vehicle's brakes. It is closely related to the coefficient of friction between the tires and the road, and on a level road the two are usually treated as the same.

The ranges in the table are wide because surfaces vary so much. Investigators therefore do not rely on tables when they can measure. The best method is a test skid or test stop: a vehicle like the one involved, on the same stretch of road in the same conditions, braked hard from a known speed while its stopping distance is measured. Investigators may also use a drag sled, a weighted object with a tire tread surface pulled across the road with a force gauge, or an accelerometer in the test vehicle that measures the deceleration directly.

From physics to a formula

The formula comes from a basic equation of motion. A vehicle that slows from speed v to a stop at a constant deceleration a covers a distance d given by v squared equals 2 times a times d. Because the deceleration is the drag factor times g, the speed at the start of braking is the square root of 2 times f times g times d. Investigators usually use a version of this formula that gives the answer directly in the units they work with.

In metric units, with d in metres and the speed in kilometres per hour, the constant 2 times 9.81, multiplied by the square of 3.6 to convert metres per second to kilometres per hour, is about 254. The working formula is the square root of 254 times f times d.

Teague points out why the test vehicle mattered. 'Anti-lock brakes stop a vehicle close to the peak grip of the tires, which is a little better than a fully locked skid on most dry surfaces. If we had tested with an old car without anti-lock brakes, we would have measured a lower drag factor and underestimated his speed. Testing with the same kind of truck gives the fairest number.'

Speed from marks to a stop

The simplest case is a vehicle that braked and slid to a stop without hitting anything. Then the formula gives its speed at the start of the marks directly. This is often used to check witness estimates and drivers' statements, and it is a common question on police training exams.

Speed when the vehicle hit something

Halloran's truck did not slide to a stop; it was still moving when it struck the hatchback. The marks show only the speed it lost while braking, not all of its speed. If the impact speed is known, the speed at the start of braking can be found by combining the two. The energy at the start of the marks equals the energy lost in braking plus the energy left at impact, and because energy depends on the square of speed, the speeds combine as the square root of the sum of their squares.

If the impact speed is not known, investigators can still give a minimum speed: the speed the vehicle would have needed just to make the marks and stop. That is the speed lost in the marks alone, and because the truck was still moving at impact, its real speed must have been higher. Minimum speeds are valuable in court because they are conservative; they do not depend on any estimate of the impact.

Uncertainty and ranges

Every measurement has some uncertainty, and investigators show the effect of that uncertainty on their conclusions. The usual approach is to calculate a range using the lowest and highest reasonable values of each input. If even the lowest value supports the conclusion, the conclusion is strong. In court, a reconstructionist who gives a range, and explains how it was calculated, is far more persuasive than one who gives a single number with false precision.

Kowalczyk's report gave the pickup's speed at the start of braking as 75 to 78 km/h, and at least 65 km/h from the marks alone. The posted limit was 50 km/h. Even the most conservative figure, the minimum from the marks alone with the lowest drag factor, was well over the limit, and that was before allowing for any slowing before the anti-lock system engaged. Teague notes how this matched Coutu's lidar reading of 81 km/h fifty-three minutes earlier, less than two kilometres away. 'Two independent methods, one an hour apart, both say the same thing about how he was driving that night.'

Grades and changes in surface

Two adjustments are common. On a hill, gravity helps or hinders the braking. On an uphill grade, the effective drag factor is the measured drag factor plus the grade, written as a decimal; on a downhill grade, the grade is subtracted. A downgrade of 5 in 100 reduces a drag factor of 0.72 to about 0.67. Kingsway is nearly level at the collision site, so no adjustment was needed.

When a vehicle slides across two surfaces, such as from asphalt onto a gravel shoulder, each section is calculated separately and the results combined, again by adding the squares. The combined formula is the square root of the constant times the sum of each drag factor multiplied by its distance.

Limits of the method

Speed from marks is reliable when the inputs are reliable, and investigators are careful about its limits. The marks must belong to the vehicle in question, which is why tire widths, positions and patterns are measured. All the tires that braked must be accounted for; a vehicle braking on only some wheels, because of a defect or a trailer, has a lower drag factor. The marks show the speed only from the point where they begin, so any slowing before they start, during the driver's reaction or before the anti-lock system engaged, is not included, which again makes the estimate conservative. And anti-lock marks can fade so that their true start is hard to see, which is why Kowalczyk's careful search for the faint start of the left front mark mattered: the 1.5 metres he found added nearly two kilometres an hour to the final estimate.

Desjardins closes the lesson with a reminder about the purpose of the arithmetic. 'Rosa Pereira's family will not care about square roots. They will care whether the man who hit their mother was speeding. The square roots are how we answer that honestly, with a number that can be checked by anyone who doubts it.'

Checking the arithmetic

Reconstructionists check every result against common sense before they rely on it. A speed that seems far too high or too low usually signals an error in a unit conversion, a misread measurement or the wrong formula. Teague's habit is to compare each answer with something she knows: a car at 50 km/h needs roughly 14 metres to stop on dry asphalt once the brakes are on, and one at 100 km/h needs roughly four times as far. If a calculation does not fit those landmarks, she looks for the mistake before writing a word of the report.