Add each part of a trip as distance and time, speed and time, or distance and speed. You get the true average speed (total distance ÷ total time) and the average velocity (displacement ÷ time), with the working shown.
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Each segment is given as
Motion
Two numbers per line in the order chosen above. Straight line: a negative distance or speed means travel back. 2-D: add a third number, the heading in degrees (0 = north, 90 = east). Use 0 speed with a time for a stop.
Enter your values to see the result.
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How to use the Average Velocity Calculator
Choose how you know each segment: distance and time, speed and time, or distance and speed.
Pick straight-line or 2-D motion. In 2-D, add a compass heading as a third number on each line.
Type one segment per line, for example 60, 60 then -60, 40 for a round trip.
Set the units for distance, time and speed.
Press Calculate and read the average speed, average velocity and the step-by-step working.
Average speed vs average velocity
Average speed = total distance ÷ total time = (d₁ + d₂ + …) ÷ (t₁ + t₂ + …). It is a scalar and is never negative.
Average velocity = displacement ÷ total time = Δx ÷ Δt. It is a vector: it has a size and a direction, and it depends only on where you start and finish.
Because distance can never be less than the size of the displacement, average speed is always greater than or equal to the size of average velocity. They are equal only when you travel in one straight line without turning back.
The harmonic mean trap
For segments of equal distance at speeds v₁ … vₙ, the average speed is the harmonic mean, n ÷ (1/v₁ + … + 1/vₙ). For segments of equal time, it is the ordinary arithmetic mean. In general, it is a time-weighted mean of the speeds, which is exactly total distance ÷ total time.
Equal-distance legs
Simple mean (wrong)
True average speed
60 and 40 km/h
50 km/h
48 km/h
30 and 60 mph
45 mph
40 mph
10 and 90 km/h
50 km/h
18 km/h
50, 60 and 75 mph
61.67 mph
60 mph
5 and 15 m/s
10 m/s
7.5 m/s
A classic puzzle: if you drive the first half of a trip at 30 mph, no speed on the second half can bring the average up to 60 mph, because the first half alone already used all the time that a 60 mph average would allow.
Worked 2-D example
Walk 3 km north in 45 minutes, then 4 km east in 1 hour. Distance = 7 km, time = 1.75 h, so average speed = 4 km/h. The displacement is √(3² + 4²) = 5 km toward a bearing of atan2(4, 3) ≈ 53.13° (northeast), so average velocity = 5 ÷ 1.75 ≈ 2.86 km/h at 53.13°.
Frequently asked questions
What is the difference between average speed and average velocity?
Average speed is total distance traveled divided by total time and has no direction. Average velocity is displacement (the straight-line change in position from start to finish) divided by total time, and it has a direction. On a round trip that ends where it started, average velocity is zero even though average speed is not.
What is the formula for average velocity?
Average velocity = Δx ÷ Δt = (final position − initial position) ÷ total time. For several segments, add the displacements as vectors (signed values on a straight line, or east and north components in 2-D) and divide by the sum of all segment times.
Why can't I just average the speeds?
Because you spend more time at the slower speed. Going 60 km at 60 km/h takes 1 hour and returning at 40 km/h takes 1.5 hours, so the average speed is 120 km ÷ 2.5 h = 48 km/h, not 50 km/h. The simple mean is only correct when every segment lasts the same amount of time.
When is average speed the harmonic mean of the speeds?
When every segment covers the same distance. Then average speed = n ÷ (1/v₁ + 1/v₂ + … + 1/vₙ). For two equal legs this is 2·v₁·v₂ ÷ (v₁ + v₂); for 60 and 40 km/h that gives 48 km/h.
Can average velocity be negative?
Yes. On a straight line the sign shows direction. If you end up behind your starting point (in the negative direction), the displacement and the average velocity are negative. Average speed is never negative.
How do I include a stop in the average?
Add a segment with zero distance (or zero speed) and the time you were stopped. The stop adds time but no distance, so it lowers both the average speed and the average velocity.
How are compass headings used in 2-D mode?
Each segment's distance is split into an east part (d·sin heading) and a north part (d·cos heading), with 0° = north and 90° = east. The parts are summed, and the displacement is √(east² + north²), with direction atan2(east, north).
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