Velocity Calculator
Calculate velocity, distance or time from the other two values.
About
Velocity Calculator
Average velocity is v = d ÷ t: distance divided by the time taken. Choose which to solve for above and the calculator rearranges the equation and converts between metres per second, kilometres per hour, miles per hour, feet per second and knots.
Speed and velocity are not quite the same
In everyday use the words are interchangeable. In physics they are not. Speed is how fast something moves. Velocity is how fast it moves in a stated direction, which makes it a vector.
The distinction has a practical consequence. Run a full lap of a 400 m track in 50 seconds and your average speed is 8 m/s, but your average velocity is zero, because you finished where you started and your displacement was nothing. Whenever a problem involves direction, circular motion or a return journey, the difference matters.
A worked example
Take 100 metres covered in 9.58 seconds, the men's world record for the distance.
- Divide distance by time: 100 / 9.58 = 10.4384 m/s
- In kilometres per hour: 10.4384 × 3.6 = 37.578 km/h
- In miles per hour: 23.350 mph
That is an average across the whole run. The peak speed during the race was higher, around 12.3 m/s, because the first two seconds are spent accelerating from a standing start. Average velocity always hides what happened in between.
Converting between units
The conversion from metres per second to kilometres per hour is a factor of 3.6, and it is worth committing to memory because it comes up constantly. It arises because there are 3600 seconds in an hour and 1000 metres in a kilometre, and 3600/1000 is 3.6.
| Unit | In m/s | Used for |
|---|---|---|
| m/s | 1 | the SI unit, used in science |
| km/h | 0.2778 | road speeds in most countries |
| mph | 0.44704 | road speeds in the US and UK |
| ft/s | 0.3048 | US engineering |
| knot | 0.51444 | sea and air navigation |
The knot is one nautical mile per hour, and a nautical mile was originally one minute of latitude, which is why it survives in navigation: it maps directly onto a chart.
Average velocity hides the detail
A journey of 120 km completed in 2 hours gives an average of 60 km/h. That is true whether the trip was a steady 60 the whole way or an hour at 100 followed by an hour at 20. The average tells you about the whole, and nothing about the parts.
This causes a classic error. Drive somewhere at 40 km/h and back at 60 km/h and the average for the round trip is not 50. Each leg covers the same distance but the slow leg takes longer, so it carries more weight. The true average is the harmonic mean, 2 × 40 × 60 / (40 + 60) = 48 km/h.
Relative velocity
Velocity is always measured against something. Two trains passing in opposite directions at 100 km/h each approach at 200 km/h relative to one another, which is why the closing speed is what matters for a collision rather than either individual figure.
The same applies to aircraft and wind. An aircraft with an airspeed of 800 km/h flying into a 100 km/h headwind covers ground at 700 km/h, and the fuel is burned against the air rather than the ground. Groundspeed determines arrival time; airspeed determines whether the wings work.
Where it gets used
Travel planning. Estimated arrival times are this equation with distance and expected average speed.
Sport. Pace, split times and race predictions are all distance over time, usually expressed as time per unit distance rather than the other way round.
Traffic engineering. Journey time reliability, junction capacity and signal timing all rest on average speeds over defined lengths of road.
Astronomy. Distances are so large that the equation is usually run backwards, using a known speed such as light to convert a travel time into a distance.
Turning velocity into pace
Runners, cyclists and swimmers rarely think in metres per second. They think in pace, which is time per unit distance, the reciprocal of speed.
A runner covering 10 km in 50 minutes has a speed of 10,000 / 3000 = 3.33 m/s, or 12 km/h. As a pace that is 5:00 per kilometre, or 8:03 per mile. Pace is more useful during an effort because it can be compared against a target without arithmetic, and most sports watches display it for that reason.
| Speed | Pace per km | 10 km time |
|---|---|---|
| 10 km/h | 6:00 | 1:00:00 |
| 12 km/h | 5:00 | 50:00 |
| 15 km/h | 4:00 | 40:00 |
| 20 km/h | 3:00 | 30:00 |
Constant velocity is rarer than it looks
The equation assumes velocity holds steady, or that an average is acceptable. Real motion rarely obliges. A journey includes acceleration from rest, stops, and a cruise that is itself uneven.
Where velocity changes at a steady rate, the average is simply the midpoint of the start and end values. Accelerating evenly from 0 to 30 m/s gives an average of 15 m/s, so the distance covered in 10 seconds is 150 m. That shortcut only works for uniform acceleration, which is why it appears throughout introductory mechanics and rarely in traffic.
Terminal velocity
An object falling through air accelerates until drag matches its weight, after which the velocity stops rising. That steady figure is terminal velocity, and it depends on mass, shape and air density rather than on how far the object has fallen.
A skydiver in a spread position reaches about 53 m/s, near 190 km/h, after roughly 12 seconds and 450 m. Head down, the smaller frontal area pushes that beyond 90 m/s. A raindrop, being small and light, tops out near 9 m/s, which is why rain does not injure anyone despite falling from a great height.
Reading a distance-time graph
Plot distance against time and the gradient of the line is the velocity. A straight line means steady motion, a steeper line means faster, and a flat line means stopped.
A curve means the velocity is changing, and its steepness at any point gives the velocity at that instant. This is why the graph is worth sketching for any problem involving a journey with stages: the shape shows immediately where the fast and slow sections were, which a single average figure hides completely.
Escape velocity and other named speeds
Some velocities are constants of a situation rather than something you choose. Escape velocity is the speed at which an object leaves a body's gravity without further propulsion: 11.2 km/s for Earth, 2.38 km/s for the Moon and 617.5 km/s for the Sun.
The speed of sound is another, though it depends on the medium: about 343 m/s in air at 20 °C, 1481 m/s in water and 5120 m/s in steel. Sound travels faster in denser, stiffer materials, which is why putting an ear to a rail detects a train long before the air carries the noise.
Common mistakes
Averaging two speeds directly. For equal distances at different speeds, use the harmonic mean rather than the arithmetic one.
Mixing time units. Distance in kilometres with time in minutes gives kilometres per minute, not per hour. Convert first.
Treating average as instantaneous. An average of 60 km/h over a journey says nothing about the fastest moment.
Ignoring direction. For a return trip the displacement is zero, so the average velocity is zero even though the average speed is not.
Common questions
Frequently asked questions
Velocity equals distance divided by time. Covering 100 metres in 9.58 seconds gives 10.4384 m/s, which is 37.58 km/h.
Speed is how fast something moves. Velocity includes direction. Run a full lap and your average speed is high while your average velocity is zero, because you end where you began.
Multiply by 3.6. There are 3600 seconds in an hour and 1000 metres in a kilometre, and 3600 divided by 1000 is 3.6.
48 km/h, not 50. The slower leg takes longer so it carries more weight. Use the harmonic mean: 2 times 40 times 60, divided by 100.
One nautical mile per hour, about 0.514 m/s. A nautical mile was originally one minute of latitude, which is why it survives in navigation.
Divide distance by speed. A 300 km journey at 100 km/h takes 3 hours.
Velocity measured against another moving object. Two trains approaching each other at 100 km/h each close at 200 km/h.
No. It gives the average across the whole distance. A sprinter averaging 10.44 m/s over 100 m peaks nearer 12.3 m/s, because the start is spent accelerating.