Displacement Integral Calculator

Calculator Inputs

Calculation mode

Lower time limit

Upper time limit

Initial position

Distance unit label

Time unit label

Coefficient a5

Coefficient a4

Coefficient a3

Coefficient a2

Coefficient a1

Coefficient a0

Sampling points for distance estimate

Sampled velocity-time pairs

Enter one pair per line using time and velocity. Commas, spaces, or semicolons are accepted.

Example Data Table

This sample illustrates how sampled velocity values can create both signed displacement and total distance estimates over a four-second interval.

Time (s)	Velocity (m/s)
0	3
1	5
2	4
3	-1
4	-2

Expected sampled-mode results for this table: signed displacement = 8.5 m, total distance = 11.7 m, average velocity = 2.125 m/s.

Formula Used

Polynomial mode: If velocity is defined as v(t), signed displacement over an interval is the definite integral s = ∫ v(t) dt from the lower time limit to the upper time limit.

For a polynomial velocity model v(t) = a₅t⁵ + a₄t⁴ + a₃t³ + a₂t² + a₁t + a₀, the calculator uses:

s = Σ [a_n / (n + 1)] × (t_upperⁿ⁺¹ − t_lowerⁿ⁺¹)

Sampled data mode: When only time and velocity samples are available, the calculator applies the trapezoidal rule:

s ≈ Σ [(v_i + v_i+1) / 2] × (t_i+1 − t_i)

Average velocity equals signed displacement divided by interval duration. Ending position equals initial position plus signed displacement. Total distance integrates the absolute value of velocity.

How to Use This Calculator

Select either polynomial velocity mode or sampled data mode.
Enter the lower and upper time limits for the motion interval.
Add an initial position if you want an ending position result.
For polynomial mode, fill in coefficients a5 through a0.
For sampled mode, paste one time-velocity pair per line.
Set unit labels so the result table matches your problem.
Press the calculate button to show results above the form.
Use the CSV or PDF buttons to download the result table.

Frequently Asked Questions

1. What does this calculator measure?

It measures signed displacement over a selected interval by integrating velocity. It also estimates total distance traveled, average velocity, peak speed within the interval, and ending position from the initial position.

2. What is the difference between displacement and distance?

Displacement keeps the sign of motion and shows net change in position. Distance ignores direction and sums how much path length was traveled, so it can be larger than displacement.

3. When should I use polynomial mode?

Use polynomial mode when velocity is described by an algebraic expression such as 0.5t² − 2t + 3. The calculator integrates that model exactly for signed displacement.

4. When should I use sampled data mode?

Use sampled data mode when you only have measured time and velocity pairs from experiments, sensors, or worksheets. The calculator then applies a trapezoidal numerical integration method.

5. Why can total distance be approximate?

In polynomial mode, distance is estimated by sampling the absolute velocity curve across the interval. Signed displacement stays exact, but distance depends on the selected sampling density.

6. Can I use custom units?

Yes. You can type any distance and time labels, including meters and seconds, kilometers and hours, or feet and minutes. The labels appear in the results automatically.

7. What happens if velocity becomes negative?

Negative velocity means motion reverses direction relative to the chosen axis. That reduces signed displacement, while total distance still counts the traveled path even during reverse motion.

8. Can I download my results?

Yes. After calculation, use the CSV button for spreadsheet-friendly output or the PDF button for a clean summary that can be saved, shared, or printed.