Velocity Calculator

What the velocity calculator does

You want to know how fast something is moving. This works it out three different ways, depending on what you already know. If you have a distance and a time, it gives you the velocity directly. If you have a starting speed, an acceleration, and a time, it gives you the velocity at the end. And if a trip came in stages at different speeds, it gives you the average over the whole thing.

Pick the method that matches your numbers and the calculator does the rest, with the answer shown in whatever speed unit you like. Below is what velocity really means, the formulas behind each method, and a worked example.

How to use it

1. Choose a method. Distance covered, acceleration, or average velocity from up to ten separate legs.
2. Enter your values. The fields change to suit the method, and each one has a unit next to it that you can change.
3. Read the result, then switch the result's unit if you want it in something other than metres per second.
4. Press Calculate for the answer, or Reset to clear it.

Speed and velocity are not the same thing

In everyday talk the two words mean the same. In physics they do not, and the difference is worth a moment. Speed is how fast you are going, a plain number like 30 metres per second. Velocity is how fast you are going and in which direction. That makes speed a scalar and velocity a vector.

The distinction has real teeth. Drive once around a track and stop where you began, and your average speed was healthy, but your average velocity is exactly zero, because your start and end positions are the same so you have gone nowhere overall. Velocity tracks the change in position, the straight line from start to finish, not the length of the path you took to get there.

This calculator works with the sizes of these quantities. When you divide a straight-line distance by time you get velocity, and when the path bends, that same division gives you average speed. For motion in a straight line, which is most everyday cases, the two are the same number.

The three ways it finds velocity

The first method is the definition of velocity itself, distance covered divided by the time it took:

v = d ÷ t

The second uses constant acceleration. If something starts at a speed u and speeds up steadily at a rate a for a time t, its final velocity is the start plus everything acceleration added along the way:

v = u + a t

The third handles a journey made of several legs, each at its own velocity for its own stretch of time. Here a plain average would be wrong, because a velocity held for a long time should count for more than one held briefly. So each leg is weighted by its time:

v_avg = (v₁ t₁ + v₂ t₂ + …) ÷ (t₁ + t₂ + …)

That is the total distance covered divided by the total time, which is the honest meaning of average velocity over a whole trip.

Units and precision

The SI unit of velocity is the metre per second, and that is the calculator's home base: every value you type is converted to it, the sum is worked out, and the result is converted back to whatever unit you chose. You can enter and read speeds in metres per second, kilometres per hour, miles per hour, feet per second, knots, and several more, and distance and time in their own range of units, so you rarely have to convert anything by hand.

Results are carried to many significant figures, far more than most situations need, so rounding inside the tool never costs you accuracy. One honest note on meaning rather than digits: the distance-over-time method gives a true velocity only when the motion is in a straight line. If the path curves, the figure is the average speed along it, which is still a useful number, just a different one.

A worked example

Take the simplest method first. A runner covers 100 metres in 12.5 seconds, so their velocity is v = 100 ÷ 12.5 = 8 m/s.

Now the acceleration method. A cyclist starts from rest and speeds up steadily at 2 m/s² for 5 seconds. Their final velocity is v = 0 + 2 × 5 = 10 m/s. The same answer in kilometres per hour is 36, which the calculator gives you with a flick of the result's unit selector.

Questions people ask

What is the difference between speed and velocity?

Speed is how fast you move, a single number. Velocity is how fast you move in a particular direction. Speed is a scalar, velocity is a vector, and a round trip back to the start has a speed but zero overall velocity.

Is this average velocity or instantaneous velocity?

It is average velocity, the value over a whole distance or time. Instantaneous velocity is the value at a single moment, which is what a speedometer shows.

Can velocity be negative?

Yes. A negative velocity simply means motion in the direction you have chosen to call negative. The size of the number is the speed, and the sign is the direction.

How do I average several velocities?

Weight each one by how long it lasted, not by a plain average. The calculator's average-velocity method does this, adding up the distance from each leg and dividing by the total time.

References

A quick note on where the physics comes from. The definition of velocity as the rate of change of position, the distinction between speed and velocity, and the constant-acceleration equation v = u + at are standard kinematics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The metre per second and the other SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 3.1, Position, Displacement, and Average Velocity. https://openstax.org/books/university-physics-volume-1/pages/3-1-position-displacement-and-average-velocity
2. HyperPhysics, Georgia State University, Description of Motion. http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811