Read and reason
Define residual before interpreting
For each time, compute residual r = measured position − predicted position. If a model predicts 5.0 m when the data point is 5.3 m, residual is +0.3 m. If it predicts 5.6 m instead, residual is −0.3 m. The sign matters: it tells you whether the model is underpredicting or overpredicting. Do not take an absolute value before studying the pattern.
Patterns can expose a wrong model
Suppose a cart truly follows a curved position trend. A best-fit straight line may underestimate at the ends and overestimate near the middle, producing a U-shaped residual pattern. A fitted quadratic can better capture the curvature in these specific synthetic cases. However, extra model parameters always improve or maintain in-sample least-squares error, so lower error by itself is not proof of a better underlying physical law.
Units and transparency
Residuals of position are measured in meters. Residuals from a velocity graph would have m/s units. Keep an original copy of the observations. Don't secretly change data values to improve a model fit. Plot residual versus time with a labeled zero line, and explain whether large residuals occur at particular times or are scattered without an obvious pattern.
Use the comparison responsibly
Scientists may compare the root-mean-square error (RMSE), the square root of the average squared residual. The units match the measured quantity. It measures training-data fit, not necessarily predictive truth. A fair model comparison also considers the number of parameters, mechanistic plausibility, measurement uncertainty and results on unseen data. This week's simulator includes a held-out point to discourage choosing a more flexible equation solely by its in-sample RMSE.