Tangential Acceleration: When the Bend Is Not the Whole Story

Hold a steady speed round a bend and the acceleration is a = v^2/r, pointing straight at the centre. Change the speed — brake, or power out — and a second component appears along the path, and the total stops pointing at the centre. The solid inward arrow is what a = v^2/r gives you; the second solid arrow is the tangential term; the dashed arrow is the acceleration the object actually has. The dashed circle is the grip budget μg drawn at the same scale, so a resultant poking outside it is a literally true picture. μ is a setting you choose here, not a measured property of any real surface.

Tangential term a_t0.000 m/s2

Grip budget μg7.848 m/s2

Budget in use63.7 %

Fastest at this a_t18.793 m/s

Fastest at a steady speed18.793 m/s

Ratio of the two1.0000

Speed where the lean hits 45°0.000 m/s

Angular velocity ω0.333 rad/s

Angular acceleration α0.000 rad/s2

The drawingNo lean is drawn: the tangential arrow is too short to see, so the dashed resultant lies along the solid inward arrow and there is no angle to mark. Its tip falls inside the dashed grip circle.

Centripetal, inward a_c = v^2/r

5.000 m/s2

Total acceleration sqrt(a_c^2 + a_t^2)

5.000 m/s2

Angle off the radius φ

0.0 deg

What the tyres are doing

Turning dominates

Speed v15.0 m/s

Bend radius r45 m

Tangential acceleration a_t0.0 m/s2

Grip coefficient μ0.80

μ is a setting, not a measurement — no value here is a sourced coefficient for any real surface. Braking and accelerating cost the same grip: the total and the angle depend on the size of a_t, never its sign. g = 9.81 m/s2. The readouts are an instantaneous snapshot, not a trajectory.