Engineer's Notebook §2.2

Tyre friction limit

A tyre can push on the road only as hard as friction allows. The total grip force, braking and cornering combined, is capped at:

F ≤ μN
F
tyre force
N
μ
coefficient of friction
—
N
normal load
N
a
acceleration
m/s²
g
gravitational acceleration
m/s²
m
vehicle mass
kg
v
speed
m/s
r
corner radius
m
d
stopping distance
m
θ
slope angle
°
A friction circle. Its radius is the most acceleration the tyres can give. An arrow from the centre shows the combined braking and cornering demand; inside the circle the tyres grip, outside they slide. 0.5 g 1 g BRAKING CORNERING GRIP USED 71.1% LIMIT μg 0.9 g
Fig. 2.2-1Circle radius μg, drawn at 100 units per g. Grey rings mark 0.5 g and 1 g.
amax = μg0.9 g
a = √(ax² + ay²)0.64 g
d = v²/2μg43.7 m
a/amax, grip used71.1%

Worked example

Uses the values in Fig. 2.2-1. Change them and the working updates.

Given
  • μdry asphalt0.9
  • axbraking0.5 g
  • aycornering0.4 g
  • vspeed100 km/h
Find

How much of the available grip the manoeuvre uses.

Solution
amax = μg = 0.9 × 9.81 m/s² = 8.83 m/s² a = √(ax² + ay²) = √(0.5² + 0.4²) g = 0.64 g = 6.28 m/s² a/amax = 0.64 g / 0.9 g = 71.1%
Answer
a/amax = 71.1%

Within the limit. Braking hard from 100 km/h on dry asphalt, the shortest stop is 43.7 m.

Related cases

  • Shortest stopping distance

    d = v²2μg

  • Fastest speed round a flat corner

    v = √(μgr)

  • Steepest slope a parked car holds

    tan θ = μ

Assumptions
  • Friction follows the Coulomb model, with μ independent of load, speed and slip. Real tyres grip best at around 10–15% slip and lose some μ under heavier load.
  • Grip is the same in every direction, so the limit is a circle. Real tyres are closer to an ellipse.
  • The car is treated as one body on a level road with N = mg: no aerodynamic downforce.
  • Friction values are typical. The real value depends on the tyre, the surface and the temperature.
  • Stopping distance excludes the driver's reaction time.

Derivation
  1. On a level road the tyres carry the car's weight: N = mg.
  2. The friction force cannot exceed μN = μmg.
  3. Newton's second law, F = ma, then caps the acceleration in any direction: a ≤ μg.
  4. Braking and cornering forces add as vectors, so the limit is √(ax² + ay²) ≤ μg: the friction circle.

Grip used for braking is not available for cornering. That is why braking hard in a corner can make the car run wide.