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Mechanical

Acceleration along a curved path

Total acceleration from the tangential and normal components.

Ref: MEC-11
Rev: v1
Field: Mechanical
Result: |a| [m/s²]

a_{t} = \frac{dv}{dt}, a_{n} = \frac{v^{2}}{ρ}

v is speed and ρ the local radius of curvature. The normal component a_n points towards the centre of curvature, even at constant speed. The magnitude is √(a_t² + a_n²).

Going round a bend you accelerate inward because your direction changes, and also along the path if you are speeding up.

Calculator

Solve for

|a|Total accelerationComputed

m/s²

vSpeed

m/s

1100

ρRadius of curvature

51,000

a_tTangential acceleration

m/s²

010

Normal acceleration4.5 m/s²
Angle from the normal23.96 °

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Total acceleration (m/s²) current value 4.924 m/s²

vSpeedKey driver

2200

ρRadius of curvature

2.0145

a_tTangential acceleration

4.511

At constant speed the acceleration is entirely normal: 4.5 m/s² for a 15 m/s vehicle on a 50 m bend.
The normal component grows with the square of speed, so halving the speed cuts it to a quarter.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m/s²)	Local step	Change in result
v Speed	1 to 100 m/s	2	200	198	+10%	+17.8 %
ρ Radius of curvature	5 to 1,000 m	2.013	45.04	43.03	+10%	−7.53 %
a_t Tangential acceleration	0 to 10 m/s²	4.5	10.97	6.466	+10%	+1.72 %

Sweep of Speed

Speed (m/s)	Total acceleration (m/s²)
1	2
1.778	2.001
3.162	2.01
5.623	2.098
10	2.828
17.78	6.633
31.62	20.1
56.23	63.28
100	200

Sweep one variable

Vary

Total acceleration (m/s²)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

|a|

Total acceleration

Result · m/s²

What the equation gives you. Everything else feeds it.

Speed

m/s · typical 1 to 100

Speed along the path. The normal component grows with v².

Radius of curvature

m · typical 5 to 1,000

Local radius of the path. A tighter bend (smaller ρ) increases the normal acceleration.

a_t

Tangential acceleration

m/s² · typical 0 to 10

Rate of change of speed along the path. Adds to the total in quadrature.

Worked example with the values above

Normal: a_n = v²/ρ = 15² / 50 = 4.5 m/s².
Tangential: a_t = 2 m/s².
|a| = √(4.5² + 2²) = 4.924 m/s².

Assumptions and limits

Radius of curvature is known and locally constant.
Speed is positive.

Common mistakes

Using the straight-line distance to the centre instead of the radius of curvature.
Adding the components arithmetically instead of in quadrature.

Statistics and uncertainty

Speed is squared, so a 2% speed error gives about 4% error in the normal component.

Where it is used

Vehicle and aircraft manoeuvre loads.
Track and road design.

Maths you need

Circular motion: v²/r
Pythagoras: combining perpendicular components

Related formulas

Practise this formula