Skid Marks, Wet Asphalt and a Speed Allegation
At 07:40 on a wet October morning a delivery van left 38 m of continuous skid marks on a straight, level stretch of national road before stopping. The speed limit is 100 km/h. The investigating officer used a coefficient of friction of 0.80 - the book value for dry asphalt - and told the driver that the marks proved she had been "well over the limit". The driver insists her speedometer read 80 km/h. Weather records show 4 mm of rain had fallen in the previous hour and the surface was still wet; the officer's own notebook records "surface damp, standing water in gutter". The van has no ABS and all four wheels locked. Its rear tyres were inflated to 1.9 bar against a recommended 2.2 bar. The driver's phone records show she was not using the phone, but she admits she reacted slowly because the car ahead braked without warning. Take g = 9.8 m/s².
1. Calculate the speed implied by the 38 m skid under (a) the officer's dry-road value μ = 0.80 and (b) a wet-road value μ = 0.55. Does the physical evidence support the claim that the van was "well over the limit"?
Read the worked answer
The friction force is F = μmg, and it does work μmg·s in removing the van's kinetic energy:
μmg·s = ½mv² → v = √(2μgs)
The mass cancels, so the result does not depend on how heavily loaded the van was.
(a) Dry: v = √(2 × 0.80 × 9.8 × 38) = √595.8 = 24.4 m/s = 88 km/h.
(b) Wet: v = √(2 × 0.55 × 9.8 × 38) = √409.6 = 20.2 m/s = 73 km/h.
Even with an unrealistically high μ = 0.90 the answer is only 93 km/h. So no plausible choice of μ puts the van above 100 km/h, and the officer's conclusion is not supported by his own measurement - he has used a coefficient appropriate to a surface the road was not. The honest statement is "between about 73 and 88 km/h, depending on the surface", i.e. consistent with the driver's account. The assumptions to flag are: level road, all four wheels braking, no significant air resistance or engine braking, and a constant μ during the skid.
2. The officer argues that a longer skid must mean a faster van, and that the driver's slow reaction makes her more culpable. Using the same data, explain what the skid length can and cannot tell you, and calculate the distance travelled before the brakes took effect if the reaction time was 1.5 s at 24.4 m/s.
Read the worked answer
The skid length fixes only the kinetic energy the van had at the instant the wheels locked. It contains no information about what happened before that instant.
Reaction distance = v·t = 24.4 × 1.5 = 36.6 m (the van is still at full speed during this time).
Total stopping distance = 36.6 + 38 ≈ 75 m.
Compare an alert driver (say 0.7 s) at 24.4 m/s: reaction distance 17.1 m, total 55 m. So a distracted driver obeying the limit can need 20 m more road than an alert one who is speeding - the marks alone cannot separate these cases. To convict on speed you would need independent evidence of v (tachograph, CCTV timing over a measured baseline, or a crash-investigation decelerometer run on that exact surface), not a friction coefficient chosen from a textbook.
3. The rear tyres were under-inflated. In which direction does this push your estimate, and how would you actually determine μ for that stretch of road rather than assuming it?
Read the worked answer
Under-inflation changes the contact patch and the pressure distribution, and in practice reduces available grip and makes the slide less repeatable, so the true μ is likely lower than the book value. A lower μ means a lower speed for the same 38 m, so the estimate of 24.4 m/s becomes an upper bound rather than a best value. The size of the effect cannot be guessed reliably - it depends on the tyre, the tread depth and the water film depth - which is exactly why it must be measured.
Practical measurement: tow a loaded test block (or a scrap wheel) at constant speed across that surface with a force meter; the pull equals μmg, so μ = F/mg. Alternatively use a vehicle with a calibrated decelerometer, or measure the deceleration of a test vehicle from a known speed with a data logger. Repeat at several points along the stretch and quote a mean and spread; that spread, not a single textbook number, is the uncertainty on the speed estimate.