Braking distance and the wet-road decision
A delivery van is travelling at 22 m/s on a straight, level road. The driver suddenly sees a fallen tree blocking the lane 55 m ahead. The driver’s reaction time before the brakes are applied is 0.8 s. On dry asphalt the brakes can produce a constant deceleration of 7.0 m/s²; on wet asphalt the maximum deceleration falls to 4.5 m/s². The van is carrying a fragile load, so the driver wants to avoid any impact if possible. The weather is overcast and the road looks damp in patches, but the driver is unsure whether the surface should be treated as dry or wet. The tree is clearly visible and there is no alternative route. The van’s brakes are in good condition and the road is straight, so the main uncertainties are the reaction time and the available friction. The driver must decide whether the current speed is safe and, if not, what maximum speed would be safe in the dry case.
1. Calculate the total stopping distance on dry and wet asphalt, and the speed at which the van would hit the tree in the wet case.
Read the worked answer
Reaction distance = 22 × 0.8 = 17.6 m.
Dry braking distance = v²/(2a) = 484/14 = 34.6 m, so total stopping distance = 17.6 + 34.6 = 52.2 m. The van stops about 2.8 m before the tree.
Wet braking distance = 484/9 = 53.8 m, so total stopping distance = 17.6 + 53.8 = 71.4 m. The van does not stop before the tree.
In the wet case, after the reaction distance there is 55 − 17.6 = 37.4 m left for braking. Using v² = 22² − 2(4.5)(37.4) gives v² = 147.4, so the impact speed is about 12.1 m/s.
2. Determine the maximum initial speed from which the van could stop before the tree on dry asphalt, and comment on whether 22 m/s is safe.
Read the worked answer
We need the total distance to be at most 55 m:
0.8v + v²/(2 × 7.0) ≤ 55,
so v² + 11.2v − 770 ≤ 0.
The positive root is
v = (−11.2 + √(11.2² + 4 × 770))/2 = (−11.2 + 56.6)/2 = 22.7 m/s.
At 22 m/s the dry stopping distance is 52.2 m, so 22 m/s is just safe in the dry case. However, the margin is only about 2.8 m, which is small compared with reaction-time uncertainty, and the van would not stop on wet asphalt.
3. Critically evaluate the constant-deceleration model. State which assumption is most likely to make the dry-case result optimistic.
Read the worked answer
The model treats deceleration as constant and at its maximum from the instant the brakes are applied. Real brakes build up force over a short time, and the available friction depends on speed, load transfer, tyre condition, and damp patches. The dry result is therefore optimistic if full deceleration is not achieved immediately or if the driver’s reaction time exceeds 0.8 s. The wet result is less sensitive to the exact value of 4.5 m/s² than to whether the driver correctly identifies the wet patches and reacts promptly.