Cross-topic physical model selection

Choosing a physical model is the first step of any solution: first we decide what may be neglected, and only then do we calculate. The candidates almost always come from topics already studied: point mass, rigid body, ideal gas, geometric optics, wave model, classical or quantum particle. Each model has a list of assumptions, and it works as long as those assumptions hold. The practical technique is to estimate characteristic quantities and form a dimensionless ratio: for instance $L/r$ for a point mass, $a/\lambda$ for geometric optics, $\lambda/L$ for quantum behaviour. If the ratio is much greater or much less than one, the choice is unambiguous; if it is of order one, we are at the edge of the model and the description must be made more detailed. The chosen model is then tested against predictions: agreement with measurement within uncertainty supports it, a systematic discrepancy signals that the model must be changed.

A satellite of size $L = 5$ m moves in a circular orbit of radius $r = 7.0\times10^{6}$ m. May its motion be described by the point-mass model? Carry out the check with a dimensionless ratio.

Sound of frequency $f = 1000$ Hz ($v = 343$ m/s) passes through a doorway of width $a = 0.9$ m, and light of wavelength $\lambda = 550$ nm passes through the same opening. For each case decide which model is required: geometric optics (straight rays) or the wave model (diffraction). Justify with the ratio $a/\lambda$.

Decide whether to treat (a) a football $m = 0.43$ kg flying at $v = 25$ m/s and (b) an electron with kinetic energy $E = 10$ eV as classical particles. The criterion is the ratio of the de Broglie wavelength $\lambda = h/p$ to the object scale $L$ ($L \approx 0.22$ m for the ball, $L \approx 0.3$ nm for the atomic scale in a crystal).

Goal: see for yourself where geometric optics ends and the wave regime begins, and obtain a numerical decision rule.

Where to work: in a simulation only — PhET "Wave Interference" (single-slit/diffraction mode) or an equivalent. Never point a real laser into an eye or at people.

1. Set $\lambda = 500$ nm and slit width $a = 20\lambda$. Sketch the pattern behind the slit — this is the "geometric" regime (a narrow strip, almost a shadow).
2. Reduce the slit: $10\lambda$, $5\lambda$, $2\lambda$, $1\lambda$, $0.5\lambda$. For each value record the ratio $a/\lambda$ and sketch the pattern.
3. Note at which $a/\lambda$ the central maximum stops being a narrow strip and broadens noticeably.
4. State your own rule: for example, "$a/\lambda \gtrsim 20$ — geometric model, $a/\lambda \lesssim 5$ — wave model".
5. Test the rule at a second wavelength ($\lambda = 650$ nm) and record whether it holds. If not, adjust the thresholds.

Safety: work in the simulation only; if you additionally listen to a tone through a doorway, keep the volume moderate and do not disturb others.

To show that model selection is not a matter of taste but a testable decision rule based on a dimensionless ratio of characteristic quantities.

A flowchart. At the top, a box "Phenomenon and question". Three candidate branches leave it: "point mass / rigid body", "wave / geometric", "classical / quantum". Each branch contains its own ratio test with a threshold: $L/r \ll 1$, $a/\lambda$ (threshold near 20), $\lambda/L$ (threshold near 1). At the bottom there are two outcome boxes: "model works — compute" and "edge of the model — refine the description", with an arrow from the second back to the list of candidates. Alongside are three small sketches: a point on an orbit with the segment $r$; a slit of width $a$ with wavefronts of wavelength $\lambda$; a particle with an attached sinusoid of wavelength $\lambda$.

A free kick from $25\ \text{m}$: a ball of mass $m = 0{,}43\ \text{kg}$ flies at $v = 25\ \text{m/s}$ and stays in the air for about $1{,}0\ \text{s}$. The model 'point mass in a gravitational field' predicts a drop of $\frac{1}{2} g t^2 \approx 4{,}9\ \text{m}$ over that time, with weight $mg \approx 4{,}2\ \text{N}$. An estimate of the drag force $F_d \approx \frac{1}{2} \rho C_d A v^2$ with $\rho = 1{,}2\ \text{kg/m}^3$, $C_d \approx 0{,}25$ and cross-section $A \approx 0{,}038\ \text{m}^2$ gives $F_d \approx 3{,}6\ \text{N}$ — the same order of magnitude as $mg$.

This is an exercise in model selection: by comparing $F_d$ and $mg$ we test the assumption of the ideal mechanical model 'a point mass with no medium resistance' and decide whether point mechanics suffices or aerodynamics with the Magnus force from the ball's spin is needed. The selection criterion is the ratio of the characteristic forces and the acceptable error in the predicted trajectory.

A kettle of power $P = 1{,}5\ \text{kW}$ heats $m = 1{,}0\ \text{kg}$ of water from $20\ ^\circ\text{C}$ to $100\ ^\circ\text{C}$ with $c = 4200\ \text{J/(kg}\cdot\text{K)}$. An energy balance without losses gives $t = \frac{m c \Delta T}{P} = \frac{1{,}0 \cdot 4200 \cdot 80}{1500} \approx 224\ \text{s}$. A stopwatch shows about $250\ \text{s}$, roughly $10\ \%$ more: the average absorbed power is $1344\ \text{W}$, and the difference $P - 1344\ \text{W} \approx 156\ \text{W}$ goes to the surroundings.

A choice is made between two thermal models: a closed system with perfect insulation ($Q_{\text{el}} = Q_{\text{heating}}$) and a system with heat losses. The discrepancy between $224\ \text{s}$ and $250\ \text{s}$ is an experimental test of the no-loss assumption; the power difference estimates whether a model with losses is required.

A phone receives a Wi-Fi signal at frequency $f = 2{,}4\ \text{GHz}$. The wavelength is $\lambda = \frac{c}{f} = \frac{3\cdot 10^8}{2{,}4\cdot 10^9} = 0{,}125\ \text{m}$. A room $5\ \text{m}$ wide is $40\lambda$, so propagation inside the room is well described by the geometric (ray) approximation. A doorway of $0{,}9\ \text{m}$ is only $7\lambda$, and diffraction at its edges is noticeable; a wall $0{,}25\ \text{m}$ thick is $2\lambda$, so the transmitted wave is strongly attenuated.

One must choose between the ray model (straight-line propagation, reflection) and the wave model (diffraction, interference). The criterion is the ratio of the obstacle or aperture size $a$ to the wavelength $\lambda$: for $a \gg \lambda$ the ray works, for $a \sim \lambda$ the wave picture does. Here $\lambda$ is obtained from $v = \lambda f$ for an electromagnetic wave with speed $c$.