Testing a simplified motion model

A simplified motion model replaces the body by a point particle and predicts the coordinate with

$$x_{\text{model}}(t)=x_0+v_0t+\frac{1}{2}at^2,$$

assuming constant acceleration and neglecting air resistance.

Video tracking or sensor data give pairs $(t_i,x_i)$ with uncertainty $\sigma_x$, so the model cannot be judged by eye — you must compute the residuals $r_i=x_i-x_{\text{model}}(t_i)$.

If $|r_i|$ are comparable to $\sigma_x$ and show no trend, the model is adequate within the measurement resolution.

A systematic trend points to a violated assumption: a residual growing like $t$ is a time-offset error, while one growing like $t^3$ indicates weak linear air drag.

The test ends with a decision — accept, refine, or reject the model — and a statement of the next measurable hypothesis.

A constant-velocity model is given by $x_{\text{model}}(t)=0.50+1.20\,t$ (SI units). Video tracking produced the coordinates $x$ = 0.52, 1.68, 2.82, 3.94 m at $t$ = 0, 1, 2, 3 s. The measurement uncertainty is $\sigma_x=0.03$ m. Compute the residuals and decide whether the model agrees with the data.

A cart rolls down an inclined track. Data: $t$ = 0, 0.2, 0.4, 0.6, 0.8 s; $x$ = 0.000, 0.038, 0.150, 0.336, 0.596 m; $\sigma_x=0.005$ m. Estimate the acceleration from the second differences, build the model $x=\frac{1}{2}at^2$ (with $x_0=0$, $v_0\approx 0$), compute the residuals, and test the model.

A ball is released from rest, and the model is $x_{\text{model}}(t)=\frac{1}{2}gt^2$ with $g=9.8$ m/s². The residuals are $r$ = $-0.001$, $-0.008$, $-0.027$, $-0.064$ m at $t$ = 0.2, 0.4, 0.6, 0.8 s; $\sigma_x=0.001$ m. A classmate claims, “This is just a time-offset error.” Test that claim by checking how the residual scales with $t$, $t^2$, and $t^3$; decide which mechanism is supported, and state a refined hypothesis together with one measurement that would test it.

Work under safe conditions, and prefer simulation whenever the measurement would require height or fast objects.

1. Choose a model: constant velocity for a toy car on a smooth floor, or uniform acceleration for a ball rolling down a low track (height no more than 10 cm); or use a simulation (PhET, ready-made Tracker examples).
2. Record video with a phone perpendicular to the plane of motion, place a ruler in the frame for scale, and extract pairs $(t_i,x_i)$ with a step of 0.2 s.
3. In a spreadsheet compute $x_{\text{model}}(t_i)$ and the residuals $r_i=x_i-x_{\text{model}}(t_i)$, then plot $r$ versus $t$.
4. Compare $|r_i|$ with $\sigma_x$ and identify the trend type: random, $\propto t$, or $\propto t^3$.
5. Record your decision: accept the model, refine an assumption (for example, add drag), or reject it, and propose the next measurable test.

Do not drop heavy objects, do not use stairs, and do not take measurements near moving traffic.

Show that the adequacy of a simplified model is judged by the structure of the residuals, not by the size of a single discrepancy, and that the shape of the trend points to a specific violated assumption.

A two-panel diagram. Top: a graph of $x$ versus $t$ with the data points and the model curve $x_{\text{model}}(t)=x_0+v_0t+\frac{1}{2}at^2$. Bottom: a graph of the residuals $r_i=x_i-x_{\text{model}}(t_i)$ versus $t$ with a $\pm\sigma_x$ band, showing three schematic cases from left to right: random scatter (model adequate), linear growth $r\propto t$ (time offset), and growth $r\propto t^3$ (air drag). Each case is labelled with the corresponding mechanism.

A phone app records the runner's time at the 20, 40, 60, 80 and 100 m marks: 2.9 s, 5.0 s, 7.0 s, 9.0 s, 10.5 s. A simple model assumes a constant speed $v = 10.0\ \text{m/s}$, i.e. $s = vt$, and predicts 2.0, 4.0, 6.0, 8.0 and 10.0 s.

The measured-minus-model differences (residuals) are $+0.9$, $+1.0$, $+1.0$, $+1.0$ and $+0.5$ s — the sign stays the same instead of scattering, so the model is systematically wrong at the start. That points to the acceleration phase $s = \frac12 a t^2$: the model is refined from the data rather than discarded.

A car must cover $150\ \text{km}$ on a motorway; the model fixes a constant speed $100\ \text{km/h}$, giving $t = s/v = 1.5\ \text{h} = 90\ \text{min}$. The clock and odometer give $118\ \text{min}$, of which $12\ \text{min}$ is a fuel stop; while moving the speedometer swings between $90$ and $110\ \text{km/h}$.

The $+28\ \text{min}$ residual splits into a term the model does not contain ($12\ \text{min}$ stopped) and a $16\ \text{min}$ remainder explained by the spread of speed around $100\ \text{km/h}$. Testing the model separates a missing term from random scatter.

A ball is dropped from $h = 1.80\ \text{m}$ and a phone films the fall at $240\ \text{frames/s}$, so consecutive frames are $1/240\ \text{s} \approx 4.2\ \text{ms}$ apart. The free-fall model without air drag, $h = \frac12 g t^2$, gives $t = \sqrt{2h/g} = \sqrt{2\cdot 1.80/9.81} = 0.606\ \text{s}$. The frames give $0.63 \pm 0.01\ \text{s}$.

The $0.024\ \text{s}$ residual (about $4\%$) exceeds the frame interval and the uncertainty in the release instant, so the no-drag model fails the test and needs a drag term ($\sim v^2$). This is where the video-based motion estimate and its time resolution enter.