M019 · Forces & measurement

Terminal Fall & Drag Laws

Can two falling masses reveal the drag law? Yes. Drop two bodies of the same size and shape but different masses; each settles at the speed where drag balances its weight. If drag grows in proportion to speed, doubling the mass doubles that speed. If drag grows with speed squared, the speed rises only by about 1.4 times.

Drop two bodies with the same cross-sectional area but different masses. Compare linear and quadratic drag, start a fall above terminal speed and estimate the 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 $\text{Ready}$

Infer the drag law from a recorded fall

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, and the second mass is a multiple of the first that you choose. Gravity stays uniform and the air stays still, at constant density, throughout a fall with no end. Ground impact, buoyancy, changing density, spin and parachute deployment are left out.

Choose vacuum, linear drag with a given coefficient, or quadratic drag with a constant coefficient. The linear coefficient does not depend on area or density. These are alternative empirical laws, not a Reynolds-number solver, and no single coefficient is claimed to hold at all speeds. Both bodies use the same law and the same coefficient.

The gold and green dashed speed curves are analytic model references. Acquire records only timestamps and the two downward speeds, with a bounded, deterministic error; no acceleration or drag is hidden in the record. The errors are not calibrated random uncertainties, and when the true speed is near zero a noisy reading can be negative.

Inference uses only the raw speeds, the known masses and gravity. Acceleration is estimated from a symmetric local slope, and endpoints and gaps are skipped. The linear and square laws are each fitted by least squares through the origin, with both bodies' data fitted together, and the inferred force keeps its sign. Noise or window bias can give negative coefficients. A smaller residual only compares the two laws; it does not single out one law or validate it experimentally.

Changing the derivative window or the replay clock keeps the raw data; changing the motion or sensor settings clears it. A wider window can wipe out real transients. The full-run work residual compares the drag work, found independently with a 600-interval trapezoid integral, with the exact change in mechanical energy. The export holds raw simulated speed data. Nothing plays automatically.

The mechanics are ideal. The page states exactly how contact is handled and what the sensors measure.

Start with the essentials

Focus question: Can two falling masses reveal the drag law?
One-sentence intuition: At terminal speed the forces balance; a body released faster than that slows down.

Core mathematical model

Equation of motion with downward speed positive

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

Both drag laws oppose the downward velocity. A body falling faster than terminal speed can have a negative downward acceleration.

Two drag laws to compare

\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 set independently. The quadratic coefficient is constant in this model.

Drag inferred from raw data

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

The local slope of speed comes from the raw times and speeds only. The reference trajectory is never used.

Work and energy

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

This integral is sampled separately, so it keeps the error that comes from its finite spacing.

Common difficulties

One plateau does not fix the drag law

Typical misconceptionOnce I see the terminal speed, I know the whole drag law.

Better mental modelA short plateau may fit several laws. Comparing two masses and recording a wide transient help. Speed noise, the derivative window and the assumption of a constant coefficient can all bias the inferred coefficients.

Run the experiment

01
Compare how mass matters
Switch between linear, quadratic and vacuum at the same mass ratio.
What to observe: Terminal speed depends on mass differently under each law. In vacuum, both bodies have the same acceleration.
02
Start faster than terminal speed
Choose Above terminal speed, then drag a sphere to replay its fall.
What to observe: Drag is larger than the weight, so the downward speed decreases.
03
Record data before fitting
Acquire a broad transient and compare the residuals of both fits.
What to observe: The scatter in force comes from differentiating the recorded speeds.
04
Test the estimate
Choose Noisy differentiation and widen the derivative window.
What to observe: The noise drops, but smoothing over the transient biases the result.

Core mathematical model

Equation of motion with downward speed positive

Two drag laws to compare

Drag inferred from raw data

Work and energy

Common difficulties

One plateau does not fix the drag law

Run the experiment

Compare how mass matters

Start faster than terminal speed

Record data before fitting

Test the estimate