Displacement Calculator

What the displacement calculator does

Displacement is how far an object ends up from where it started. This calculator works it out from whatever you know about the motion, using four different methods. You can give it an average velocity and a time, or a starting velocity with an acceleration, or a starting and final velocity, or a string of separate velocities for a trip in stages.

Each method is a different route to the same quantity. Below is how displacement differs from plain distance, the formula behind each method, and a worked example.

How to use it

1. Choose a method. Average velocity, initial velocity and acceleration, initial and final velocity, or up to ten different velocities.
2. Enter the values the method asks for. The fields use SI units: metres per second, metres per second squared, and seconds.
3. Press Calculate for the displacement in metres, or Reset to clear it.

Displacement is not distance

These two get used as if they mean the same, but in physics they part ways. Distance is the total length of the path travelled, every twist and turn added up. Displacement is only the straight line from start to finish, with a direction attached. Distance is a scalar, displacement is a vector.

Walk five steps forward and five back, and you have covered a distance of ten steps but your displacement is zero, because you finished where you began. The same idea makes displacement able to be negative: a negative value just points the opposite way along your chosen direction. This calculator reports the displacement, the net change in position, which is usually what a motion problem is really asking for.

The four ways it finds displacement

The simplest method uses an average velocity held over a time. Displacement is just the two multiplied:

s = v_avg t

The second uses a starting velocity and a steady acceleration. The object covers the ground it would at its starting speed, plus the extra that acceleration piles on, which grows with the square of the time:

s = u t + ½ a t²

The third takes a starting and a final velocity over a time. Since the speed climbs steadily, its average is just the midpoint of the two, and displacement is that average across the time:

s = ½ (u + v) t

The fourth adds up a journey leg by leg, each velocity across its own time, and sums the displacements. All four trace back to one idea: displacement is the area underneath a graph of velocity against time.

Units and precision

This calculator works in SI units throughout: velocity in metres per second, acceleration in metres per second squared, time in seconds, and the displacement it returns in metres. Keeping everything in one consistent set of units is what makes the equations line up, so if your figures start out in other units, convert them first and the result will be in metres.

Results are rounded to three decimal places, which is finer than almost any real measurement of motion. The rounding is only in the displayed answer, while the calculation itself runs at full precision.

A worked example

An object starts at 5 m/s and accelerates steadily at 2 m/s² for 4 seconds. How far does it travel?

Use the second method, s = u t + ½ a t². The first part is 5 × 4 = 20 metres, the ground covered at the starting speed. The second part is ½ × 2 × 4² = 16 metres, the extra from speeding up. Together the displacement is 20 + 16 = 36 metres.

Questions people ask

What is the difference between displacement and distance?

Distance is the whole length of the path travelled. Displacement is the straight-line gap from start to finish, with a direction. A round trip has plenty of distance but zero displacement.

Can displacement be zero or negative?

Both. It is zero when you end where you started, and negative when the net movement points the opposite way along your chosen direction.

Which method should I use?

Whichever matches the numbers you have. Use average velocity and time if that is what you know, the acceleration method if you have a starting speed and acceleration, and the two-velocity method if you know the speeds at each end.

What is the displacement equation for constant acceleration?

It is s = u t + ½ a t², where u is the initial velocity, a is the acceleration, and t is the time. It is one of the standard equations of motion.

References

A quick note on where the physics comes from. The difference between displacement and distance, and the equations of motion for constant acceleration such as s = ut + ½at², are standard kinematics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The metre and the other SI units follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 3.4, Motion with Constant Acceleration. https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration
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