The Newsletter Claim: Is 8 m/s Safe on a Wet Path?
A Year 11 student is running a bike-safety investigation for the school newsletter. She rides a bike-and-rider combination of total mass 75 kg along a straight, level path at a steady 8.0 m/s and measures the distance travelled from the moment a friend shouts 'stop' until she is stationary. Her reaction time is measured separately as 0.70 s. On dry asphalt her braking distance is 6.4 m. She repeats the trial after the path has been hosed down and finds that, at the same 8.0 m/s, the braking distance is 11.2 m. She assumes the braking force is constant in each trial and that the path is level. A side driveway joins the path at a blind corner, and a pedestrian stepping out there would first be visible 15 m ahead of her. She wants to write in the newsletter that '8 m/s is a safe speed on this path'. Her teacher asks her to check that claim against her own data before it goes to print.
1. Calculate the student's total stopping distance on dry asphalt and on wet asphalt at 8.0 m/s. Use the 15 m sight line at the blind corner to judge whether her newsletter claim is safe.
Read the worked answer
Reaction distance depends only on speed and reaction time, so it is the same in both trials:
- d = v t = 8.0 x 0.70 = 5.6 m
Total stopping distance = reaction distance + braking distance:
- Dry: 5.6 + 6.4 = 12.0 m
- Wet: 5.6 + 11.2 = 16.8 m
The pedestrian first becomes visible 15 m ahead.
- Dry: she stops 3.0 m before the pedestrian. Safe at 8.0 m/s.
- Wet: she needs 16.8 m, so she is still moving at 15 m. She has used 5.6 m of reaction and 9.4 m of braking, so v^2 = 8.0^2 - 2 x 2.86 x 9.4 = 64 - 53.8 = 10.2, giving an impact speed of about 3.2 m/s (roughly 11 km/h). She hits the pedestrian.
The claim as written is not safe because it is a single number for a path whose grip changes. A defensible statement is surface-specific and leaves a margin, e.g. '8 m/s is fine on dry asphalt, but on wet asphalt you cannot stop within 15 m, so slow down before the blind corner.'
2. Use the wet-road braking distance to find the average braking force on the bike, and compare it with the dry-road force. Explain the difference in terms of forces, not just numbers.
Read the worked answer
The braking force is what produces the deceleration, so work from the kinematics first.
Wet asphalt: v^2 = 2ad with v = 8.0 m/s, d = 11.2 m
a = v^2 / (2d) = 64 / 22.4 = 2.86 m/s^2 F = ma = 75 x 2.86 = about 214 N (2.1 x 10^2 N)
Dry asphalt: d = 6.4 m
a = 64 / 12.8 = 5.0 m/s^2 F = 75 x 5.0 = 375 N
The wet braking force is only 214/375 = 0.57, about 57% of the dry force. The rider's mass and the brakes themselves have not changed, so the difference must be at the tyre-road contact. A water film partly separates the rubber from the asphalt, reducing the interlocking and adhesion that produce friction, so the maximum friction force available is smaller and the bike takes longer to slow. This also explains why the same brake lever pull gives a smaller deceleration on the wet path: the limiting factor is grip, not brake force.
Assumptions worth stating: constant deceleration, level path, no significant air resistance, and the same reaction time in both trials.
3. The student still wants one 'safe speed' for the wet path that stops her within 15 m. Calculate that speed and evaluate how much confidence the school should place in it.
Read the worked answer
Set total stopping distance on wet asphalt equal to 15 m, using the wet deceleration a = 2.86 m/s^2:
0.70v + v^2 / (2 x 2.86) = 15
0.70v + v^2 / 5.71 = 15
Multiply through by 5.71:
v^2 + 4.0v - 85.7 = 0
v = [-4.0 + sqrt(16 + 342.9)] / 2 = (-4.0 + 18.9) / 2 = 7.5 m/s (about 27 km/h)
So about 7.5 m/s is the maximum wet-path speed that just stops in 15 m.
How much confidence? Treat it as a ceiling, not a safe speed. It depends on a reaction time measured in a calm test, whereas a tired or distracted rider reacts more slowly, and every 0.1 s of extra reaction adds 0.75 m. It assumes constant deceleration on level ground, but the path slopes slightly and wet grip varies with tyre wear, tread and how much water is lying. It also assumes the brakes are perfectly applied. Real safety advice should set the limit well below 7.5 m/s and, better, remove the problem by slowing before the blind corner rather than relying on stopping after the pedestrian appears.