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Simple speed from skid marks

To calculate the speed at which a vehicle must travel in order to leave tyre marks on the road surface of a specific length the following will apply. Consider a vehicle has skidded to a stop in a straight line leaving all 4 tyre marks on the road surface, there is certain data we will need:

Some of these figures are considered standards.

For example the rate of gravity is considered to be 9.81 and is represented as (g). For non police calculations when skid tests can not be carried out and in the absence of any other information the coefficient of friction between the road and tyre might be considered .7, coefficient of friction is represented by (mu).

The accepted formulae is v2=u2 +-2mugs

Having established the following data:

skidded displacement 13m represented by (s) and came to a stop represented by (v) final velocity we can calculate that u (initial velocity) = (sq)2*.7*9.81*13 (were v = zero) which would calculate to 13 metres per second which is about 30 m.p.h.

Table of symbols

v = final velocity, u = initial velocity, mu = coefficient of friction, g = gravity, s = skidded displacement.

This basic calculation and its transformations runs through many calculations used in accident investigation and is worth getting to grips with. Note: The coefficient of friction of a new road surface can measure as low as .5 mu. This is because a new surface has granite chippings rolled into the tarmac. Tarmac has a lower coefficient of friction than granite chippings and until the tarmac is worn down to reveal the granite vehicles are skidding on more tarmac than granite.

Having started with the simple things in life, there is a complication relating to the inclines and declines of our roads. Inclines and declines effect the results of calculations made as the forces alter in such circumstances.

The coefficient of friction between road surface and tyre remains the same on sloping surfaces however the effect of gravity on the weight of the vehicle changes things quite considerably. Needless to say it will take longer for a vehicle to stop going down a slope than it will on the flat and a shorter distance on an incline. The calculation involves triangles and the relevant mathematics:

Note: ( a = mug ) (d = downhill) (f = flat) (u = uphill)

acceleration ‘a’ downhill is: ad = afcos0 - gsin0

re-arranged for aflat is: af = au - gsin0 = ad + gsin0
                                                     ------------- ------------
                                                                cos0 cos0

The figure determined from these equations is then used in place of ‘mug’ to represent the acceleration rate.

A further complication is that of unbraked trailers towed by vehicles

In an ordinary situation when a vehicle locks its wheels and skids the only deceleration value is that of the coefficient of friction between tyre and road surface, the weight or mass of the vehicle is of no account. When an unbraked trailer is involved the trailer having no brakes introduces a weight element due to the rotating wheels of the trailer. Again there is a formulae to assist the expert:

On a flat road ares = mc af
                             -------------
                                mc + mt

On a downhill incline the formulae needs to include the slope therefore the formulae changes thus:

ares = mc ad - mt gsin0
         ------------------------
                   mc + mt

On an uphill incline the positive and negative elements of the equation are reversed. The (ares) value is then used to represent the (mug) element of the equation.

As a simple guide to the likely speed of a vehicle subject to straight forward wheel lockup on a flat surface.

Acceleration Rate/Speed Chart. (all figures are an approximate guide only), skidded displacement in metres..

'm'	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
‘a’
Rate
5.5	23	24	25	26	27	28	29	30	31	32	33	34	35	36	36	37
6.0	24	25	26	27	29	30	31	32	33	34	34	35	36	37	38	39
6.5	25	26	28	29	30	31	32	33	34	35	36	37	38	39	40	40
7.0	26	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42
7.5	27	28	30	31	32	33	34	35	36	38	38	39	40	41	42	43
8.0	28	29	31	32	33	34	36	37	38	39	40	40	42	43	43	44