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Mechanical

Acceleration from the velocity gradient

Acceleration when speed is known as a function of position.

Ref: MEC-09
Rev: v1
Field: Mechanical
Result: a [m/s²]

a = \frac{dv}{dt} = v \frac{dv}{ds}

For motion along a straight axis with signed position s. Using v = ds/dt in the chain rule gives a = v dv/ds, useful when acceleration or speed is given as a function of position rather than time.

If you know how quickly speed changes per metre travelled, multiply by the speed to get the acceleration per second.

Calculator

Solve for

aAccelerationComputed

m/s²

vSpeed

m/s

1100

dv/dsVelocity gradient

1/s

−22

Distance to stop if the gradient stays constant62.5 m
Acceleration in g−1.02 g

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.

Acceleration (m/s²) current value −10 m/s²

dv/dsVelocity gradientKey driver

−5050

vSpeed

−40−0.4

A negative result is a deceleration: the vehicle is slowing by 0.4 m/s for every metre.
Fast-moving objects feel large accelerations even for gentle gradients, because the speed multiplies the gradient.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m/s²)	Local step	Change in result
dv/ds Velocity gradient	−2 to 2 1/s	−50	50	100	+10%	+10 %
v Speed	1 to 100 m/s	−40	−0.4	39.6	+10%	+10 %

Sweep of Speed

Speed (m/s)	Acceleration (m/s²)
1	−0.4
1.778	−0.7113
3.162	−1.265
5.623	−2.249
10	−4
17.78	−7.113
31.62	−12.65
56.23	−22.49
100	−40

Sweep one variable

Vary

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

Variables and how each one contributes

Acceleration

Result · m/s²

What the equation gives you. Everything else feeds it.

Speed

m/s · typical 1 to 100

Speed at the position of interest. Directly proportional.

dv/ds

Velocity gradient

1/s · typical −2 to 2

Rate of change of speed per metre travelled (negative when slowing). Directly proportional; its sign gives the sign of the acceleration.

Worked example with the values above

Speed v = 25 m/s; gradient dv/ds = −0.4 1/s.
Multiply: v × dv/ds = 25 × −0.4.
a = −10 m/s².

Assumptions and limits

Motion along a straight line with signed position.
Speed varies smoothly with position.

Common mistakes

Using the average speed instead of the speed at the point.
Dropping the sign.

Statistics and uncertainty

Both factors are measured, so their relative errors add in quadrature; a speed gradient from sampled data is usually the noisier term.

Where it is used

Braking and arrester-gear profiles.
Acceleration along a track from a speed map.

Maths you need

Chain rule: dv/dt = (dv/ds)(ds/dt)
Derivative: rate of change

Related formulas

Practise this formula