Calculator inputs
Example data table
These sample datasets help you test both modes quickly.
Position-time example
| Time (s) | Position (m) |
|---|---|
| 0 | 0.2 |
| 1 | 1.9 |
| 2 | 6.0 |
| 3 | 12.5 |
| 4 | 21.4 |
| 5 | 32.7 |
Velocity-time example
| Time (s) | Velocity (m/s) |
|---|---|
| 0 | 1.2 |
| 1 | 3.6 |
| 2 | 5.9 |
| 3 | 8.5 |
| 4 | 10.9 |
| 5 | 13.4 |
Formula used
1. Position-time trendline
The calculator fits a quadratic trendline: s(t) = at² + bt + c. When motion follows constant acceleration, the coefficient of t² controls acceleration.
The estimated acceleration becomes: Acceleration = 2a. The instantaneous velocity at any time is: v(t) = 2at + b.
2. Velocity-time trendline
The calculator fits a linear trendline: v(t) = mt + c. Here, the slope m directly represents acceleration.
The estimated acceleration becomes: Acceleration = m. This mode works well when your measured values are already velocities.
3. Fit quality
R² shows how well the trendline explains the data. RMSE shows the typical prediction error magnitude. Lower RMSE and higher R² usually indicate a better fit.
How to use this calculator
- Select Position vs time or Velocity vs time.
- Enter time values in order or let the page sort them.
- Enter the matching measured values in the second data box.
- Set units for time, measured values, and optional acceleration output.
- Choose a prediction time to estimate a fitted future value.
- Press the calculate button to show results above the form.
- Review acceleration, equation, fit statistics, residuals, and the graph.
- Export the results with the CSV or PDF buttons.
Frequently asked questions
1. What does this calculator estimate?
It estimates acceleration from fitted trendlines. Position-time data uses a quadratic fit. Velocity-time data uses a linear fit. It also reports prediction values, residuals, R², RMSE, and the fitted equation.
2. When should I use position-time mode?
Use position-time mode when your measured values are distances or displacements. The calculator fits a quadratic curve and converts its squared-time coefficient into estimated acceleration for near-constant acceleration motion.
3. When should I use velocity-time mode?
Use velocity-time mode when your measured values are velocities. The trendline slope becomes acceleration directly. This is often the simplest method when speed measurements already exist.
4. Do the time intervals need to be equal?
No. Regression fitting can handle uneven time intervals. The important requirement is that every time value matches the correct measured value in the same position.
5. What do residuals show?
Residuals show the difference between actual measurements and fitted values. Large residuals can indicate noise, outliers, poor model choice, or behavior that does not follow the selected trendline type.
6. Is a high R² always enough?
No. High R² helps, but it does not guarantee a physically meaningful model. You should also inspect RMSE, residual patterns, data quality, and whether the chosen motion model fits the real system.
7. Can I trust extrapolated predictions?
Extrapolated predictions can be useful, but they carry more uncertainty. Results are strongest inside the measured range. Outside that range, real motion may change and reduce accuracy.
8. What units should I enter?
Enter units that match your data. For example, use seconds for time, meters for position, or meters per second for velocity. You can also override the acceleration unit manually.