Average Velocity Formula

Average velocity is total displacement divided by total time: v̄ = Δx ÷ Δt. It differs from average speed, which uses total distance, whenever you change direction. Enter the legs of a journey — each with a distance, a time or speed, and a direction — and the calculator gives both averages with the working shown.

The average velocity formula

Average velocity is the change in position divided by the time taken: v̄ = Δx ÷ Δt = (x_final − x_initial) ÷ (t_final − t_initial). Because displacement has a direction, average velocity is a vector: it can be positive, negative or zero. A runner who completes one lap of a 400 m track in 80 seconds has an average speed of 5 m/s but an average velocity of zero, because they finish where they started.

Average velocity vs average speed

Constant acceleration shortcut

When acceleration is constant, average velocity is simply the mean of the initial and final velocities: v̄ = (u + v) ÷ 2. A car accelerating steadily from 10 m/s to 30 m/s has an average velocity of 20 m/s, so in 5 seconds it travels 100 m. This shortcut does not work when acceleration changes — then you need total displacement divided by total time.

Why you can’t just average the speeds

If you drive 60 km at 30 km/h and then 60 km at 90 km/h, your average speed is not 60 km/h. The first leg takes 2 hours and the second 40 minutes, so 120 km in 2 h 40 min gives 45 km/h. For equal distances, the correct average is the harmonic mean: 2 × 30 × 90 ÷ (30 + 90) = 45 km/h. Always add up total distance and total time.

How to use the calculator

1. Enter each leg on its own line: distance with units (m, km, mi, ft), then either the time (s, min, h) or the speed (m/s, km/h, mph), then + or − for direction.
2. Choose the units for the results.
3. Read total distance, displacement and time, and the average speed and average velocity.
4. Use the table to check each leg’s time and speed.

Worked example

A cyclist rides 3 km east in 10 minutes, 2 km back west in 12 minutes, then 5 km east at 60 km/h (5 minutes). Total distance is 10 km and total time 27 minutes, so the average speed is 22.2 km/h. The displacement is 3 − 2 + 5 = 6 km east, so the average velocity is 6 km ÷ 0.45 h = 13.3 km/h east.

Instantaneous vs average velocity

Instantaneous velocity is the velocity at a single moment — what a speedometer shows, with a direction. On a position–time graph, average velocity is the slope of the straight line joining two points, while instantaneous velocity is the slope of the tangent at one point. As the time interval shrinks, the average velocity approaches the instantaneous velocity, which is the idea behind the derivative in calculus.

Units of velocity

To convert km/h to m/s divide by 3.6; to convert m/s to km/h multiply by 3.6.

Velocity in two dimensions

When motion isn’t along a straight line, displacement is a vector with both size and direction. Walk 3 km north and then 4 km east and you have travelled 7 km, but your displacement is 5 km to the north-east (by Pythagoras: √(3² + 4²) = 5). If the walk took 1.4 hours, your average speed was 5 km/h and your average velocity 3.6 km/h in a direction about 53° east of north. The calculator handles journeys along one line with + and − directions.

Graphs of motion

On a position–time graph, average velocity between two times is the slope of the straight line joining the two points: rise (change in position) over run (change in time). A steeper line means a higher velocity; a downward slope means negative velocity, moving back towards the start. On a velocity–time graph, the area under the line gives the displacement, so the average velocity is that area divided by the total time.

Common mistakes

• Averaging speeds instead of dividing total distance by total time.
• Using distance instead of displacement for velocity.
• Mixing units — convert minutes to hours or seconds first.
• Forgetting that velocity needs a direction.
• Using (u + v)/2 when acceleration is not constant.

What the template pack includes

Printable physics worksheets on average velocity and speed with worked answers, a motion formula sheet, position–time graph paper and a lab data sheet for measuring walking and cycling speeds.