M019 · Forces & measurement

Terminal Fall & Drag Laws

Drop equal-area bodies with different masses, compare linear and quadratic resistance, start above terminal speed and infer drag from a sampled speed record.

Measurement state $\text{Ready}$

Selected time $\text{Ready}$

Model downward speed A $\text{Ready}$

Model downward speed B $\text{Ready}$

Reference terminal speed A $\text{Ready}$

Reference terminal speed B $\text{Ready}$

Terminal speed ratio $\text{Ready}$

Upward drag on A $\text{Ready}$

Downward acceleration A $\text{Ready}$

Raw-fit linear coefficient $\text{Ready}$

Raw-fit quadratic coefficient $\text{Ready}$

Linear force residual RMS $\text{Ready}$

Quadratic force residual RMS $\text{Ready}$

Usable inferred force points $\text{Ready}$

Full-run sampled work residual A $\text{Ready}$

Recorded sensor rows $\text{Ready}$

RELEASE / RECORD / INFER

Test the force law

Mass A

m_A

1.00\,kg

Mass ratio

\gamma

2.00\,

Recording duration

T

12.00\,s

Initial downward speed

u_0

0.00\,\frac{\mathrm m}{\mathrm s}

Drag: vacuum / linear / quadratic

L

2.00\,

Common cross-sectional area

A

0.20\,\mathrm m^2

Air density

\rho

1.23\,\frac{\mathrm{kg}}{\mathrm m^3}

Quadratic drag coefficient

C_D

1.00\,

Independent linear coefficient

b

0.50\,\frac{\mathrm{kg}}{\mathrm s}

Bound on speed error

\epsilon_u

0.02\,\frac{\mathrm m}{\mathrm s}

Odd derivative window

n_w

5.00\,

Gravity magnitude

g

9.80\,\frac{\mathrm m}{\mathrm s^2}

Sensor sampling rate

f_s

10.00\,Hz

Deterministic error seed

s

17.00\,

Replay fraction

\tau

0.40\,

Both spheres have the same cross-sectional area; the second mass is a chosen multiple of the first. Uniform gravity and still, constant-density air persist throughout an unbounded fall. There is no ground impact, buoyancy, changing density, spin or parachute deployment.

Choose vacuum, a prescribed linear coefficient, or constant-coefficient quadratic drag. The linear coefficient is independent of area and density. These are alternative phenomenological laws, not a Reynolds-number solver or a claim that one coefficient applies at all speeds. The same law and coefficient act on both bodies.

Gold and green dashed speed curves are analytic model references. Acquire records only timestamps and two downward speeds with bounded deterministic error; no acceleration or drag is secretly stored. Errors are not calibrated random uncertainties, and a near-zero true speed may yield a negative noisy reading.

Inference receives only raw speeds, known masses and gravity. A symmetric local slope estimates acceleration; missing endpoints and gaps are omitted. Joint zero-intercept linear and square-law least-squares fits retain signed inferred force. Noise or window bias can produce negative coefficients. Smaller residual is a comparison, not unique identification or experimental validation.

Changing the derivative window or replay clock preserves raw data; changing motion or sensor settings clears it. A wider window can erase genuine transients. The full-run work residual compares an independent 600-interval trapezoid drag-work integral with the exact mechanical energy change. Export is raw simulated speed data. No autoplay.

Ideal mechanics, explicit contact and measurement boundaries.

Start with the essentials

Focus question: Can two falling masses reveal the drag law?
One-sentence intuition: At terminal speed the forces balance; a superterminal release slows down.

Core mathematical model

Downward speed convention

m\dot u=mg-D(u),\qquad u=-\dot y\geq0

Both models oppose the downward velocity; overspeed can produce negative downward acceleration.

Alternative force laws

\begin{aligned}D_1&=bu,\quad u_{T,1}=mg/b\\D_2&=cu^2,\quad c=\rho C_D A/2\\u_{T,2}&=\sqrt{mg/c}\end{aligned}

The linear coefficient is independent; the quadratic coefficient is constant in this model.

Independent raw-data inference

\widehat D_i=m\left(g-\widehat{\dot u}_i\right)

A local speed slope uses raw times and speeds, without consulting the reference trajectory.

Work and energy

\Delta(K+U)=-\int_0^T D(u)u\,\mathrm dt

The separately sampled integral retains its finite-spacing error.

Common difficulties

Read the force correctly

Typical misconceptionA terminal-speed trace uniquely determines the whole drag law.

Better mental modelA short plateau may fit several laws. Mass scaling and a wide transient help, while speed noise, derivative windows and constant-coefficient assumptions can bias the inferred coefficients.

Run the experiment

01
Compare mass scaling
Switch between linear, quadratic and vacuum at the same mass ratio.
What to observe: Terminal speeds scale differently; vacuum accelerations coincide.
02
Start above equilibrium
Choose Above terminal speed, then scrub the gold clock.
What to observe: Drag exceeds weight and downward speed decreases.
03
Acquire before inferring
Acquire a broad transient and compare both fitted residuals.
What to observe: The force scatter comes from differentiated recorded speeds.
04
Challenge the estimate
Choose Noisy differentiation and widen the derivative window.
What to observe: Noise decreases, but transient smoothing introduces bias.

Core mathematical model

Downward speed convention

Alternative force laws

Independent raw-data inference

Work and energy

Common difficulties

Read the force correctly

Run the experiment

Compare mass scaling

Start above equilibrium

Acquire before inferring

Challenge the estimate