Library / Foundations: Maths / FMA-34

Foundations: Maths

Distance from average velocity

Distance travelled under constant acceleration from start and end speeds.

Ref: FMA-34
Rev: v1
Field: Foundations: Maths
Result: s [m]

s = \frac{1}{2} (u + v) t

Apply separately along each axis. u and v are the initial and final velocities and t the time. Valid for constant acceleration.

With constant acceleration the average velocity is the mean of the start and end velocities; multiply by time to get distance.

Calculator

Solve for

sDistanceComputed

uInitial velocity

m/s

030

vFinal velocity

m/s

1050

tTime

160

Average velocity10 m/s
Acceleration (v − u)/t1.25 m/s²

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.

Distance (m) current value 80 m

tTimeKey driver

10600

vFinal velocity

60220

uInitial velocity

60180

It gives distance without needing the acceleration.
The area of a trapezium under a velocity–time graph is exactly this formula.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m)	Local step	Change in result
t Time	1 to 60 s	10	600	590	+10%	+10 %
v Final velocity	10 to 50 m/s	60	220	160	+10%	+7.5 %
u Initial velocity	0 to 30 m/s	60	180	120	+10%	+2.5 %

Sweep of Time

Time (s)	Distance (m)
1	10
1.668	16.68
2.783	27.83
4.643	46.43
7.746	77.46
12.92	129.2
21.56	215.6
35.97	359.7
60	600

Sweep one variable

Vary

Distance (m)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Distance

Result · m

What the equation gives you. Everything else feeds it.

Initial velocity

m/s · typical 0 to 30

Velocity at the start. Raises distance.

Final velocity

m/s · typical 10 to 50

Velocity at the end. Raises distance.

Time

s · typical 1 to 60

Elapsed time. Directly proportional.

Worked example with the values above

Average velocity: (5 + 15)/2 = 10 m/s.
Multiply by time 8 s.
s = 80 m.

Assumptions and limits

Constant acceleration.
Straight-line motion with a chosen positive direction.

Common mistakes

Using it when acceleration changes.
Mixing km/h and m/s.

Statistics and uncertainty

Errors in u, v and t combine; timing error dominates for short intervals since t multiplies the whole result.

Where it is used

Braking and run-up distances.
Distance from a velocity–time graph.

Maths you need

Velocity: speed with direction
Average: mean of two values

Related formulas

Practise this formula