Math Bridge: Fall Motion and Sampling

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Can a 50 Hz clock catch a 20 ms fall impact?

Follow one fall from nanometre proof-mass motion to ADC codes and sample timing.

Motion Marley, the movement guideMotion Marley guides
The one targetSeparate amplitude resolution from time resolution.
The chapter case3g, 5.5 kHz resonance, ±16g, 12 bits, 20 ms, and 50 Hz.
What it buys youA testable explanation for missed impact peaks.

A field team faces an unresolved physical question: Can a 50 Hz clock catch a 20 ms fall impact? They must answer it before changing sample rate on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is sample rate. The middle card applies this page's relationship. The green card is 3g displacement. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Sample rate changes 3g displacement An input card leads through the page relationship to the 3g displacement result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Quantisation divides amplitude; sampling divides time. Passing one does not repair failure in the other.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for sample rate is 50.

  2. 2

    Name the relationship. x3g=3x9.81/(2πx5500)²=24.6 nm x16g=130.5 nm q=32g/4096=7.8125 mg/code 3g/q=384 codes; 2g/q=256 codes fpulse≈1/0.020=50 Hz fs,min=2x50=100 Hz; 50x0.020=1 sample

  3. 3

    Substitute the chapter fixture. Set sample rate to 50. The page ledger gives 3g displacement as 24.644 nm.

  4. 4

    Read the result. Keep nm beside the value. Use it only inside the technical boundary on this page.

Predict, then change sample rate

Try Predict the direction of 3g displacement. Move one control, calculate, then check your prediction.

50
Chapter baseline
3g displacement

Observe Quantisation divides amplitude; sampling divides time. Passing one does not repair failure in the other. Reset the control to 50 and compare 3g displacement.

Explain Only sample rate moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only sample rate moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only sample rate. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

A MEMS accelerometer reads the motion of a tiny spring-mounted mass. Its converter can resolve many amplitude steps, yet a slow clock can still sample before and after a brief impact. Good codes cannot recover a peak that was never sampled.

Motion Marley: Ask how finely you measure and how often you look.

2. Name every algebra move

1

Cancel the massSubstitute k=m(2πfn)² into x=ma/k.

2

Find displacementDivide acceleration by squared resonance angular frequency.

3

Find one codeDivide the full 32g span by 2¹².

4

Convert pulse to frequencyUse approximately one over 20 ms.

5

Apply NyquistDouble the fastest feature frequency.

6

Count opportunitiesMultiply sample rate by pulse duration.

3. Reproduce the chapter case

x3g=3×9.81/(2π×5500)²=24.6 nm
x16g=130.5 nm
q=32g/4096=7.8125 mg/code
3g/q=384 codes; 2g/q=256 codes
fpulse≈1/0.020=50 Hz
fs,min=2×50=100 Hz; 50×0.020=1 sample

The ADC has ample nominal code resolution for the thresholds, while 50 Hz offers only one sample opportunity in the illustrative impact window.

4. Try one real input

TryRaise sample rate and predict the sample period and opportunities inside 20 ms.

Sample rate
3g displacement
16g displacement
ADC step (mg)
3g codes
2g codes
Sample period
Pulse frequency
Nyquist minimum
Samples per pulse
Resonance/sample ratio

ObserveAt 50 Hz the sample period equals the pulse duration. At 100 Hz there are two nominal opportunities.

ExplainQuantisation divides amplitude; sampling divides time. Passing one does not repair failure in the other.

Technical boundaries.

This is a quasi-static mass-spring and pulse-width screen.

MEMS
Damping, bandwidth, axis alignment, analogue filtering, noise, saturation, and device calibration shape the reading.
Falls
Impact duration, body location, surface, activity, and person vary; 20 ms is illustrative.
Algorithm
Threshold timing alone does not establish a fall, urgency, or clinical outcome.

Correct, not complete: this calculation does not validate a fall detector or care pathway.

5. Use the result in the design

Choose rate and anti-alias filtering from measured impact spectra, then test phase, body location, activities of daily living, cancellation, and escalation with the intended population.

6. Record the evidence state

Keep sensor model, range, rate, filter, axis, mounting, calibration, timestamp, raw trace, threshold, impact width, activity label, user confirmation, alert delivery, and outcome.

7. Check yourself

Why does the proof mass move only nanometres?
Answer: Its high mechanical resonance corresponds to a stiff spring, so acceleration produces very small displacement.
Why are 384 codes not enough evidence?
Answer: They describe nominal amplitude quantisation, not whether the clock sampled the peak.
Does 100 Hz validate fall detection?
Answer: No. It clears only this simplified 20 ms Nyquist screen; filtering, noise, behavior, and outcomes still need validation.
Honesty boundary.

The arithmetic reproduces the chapter's 3g and 50 Hz case with catalog-style sensor assumptions.

MEMS
Damping, bandwidth, axis alignment, analogue filtering, noise, saturation, and device calibration shape the reading.
Falls
Impact duration, body location, surface, activity, and person vary; 20 ms is illustrative.
Algorithm
Threshold timing alone does not establish a fall, urgency, or clinical outcome.

Correct, not complete: this calculation does not validate a fall detector or care pathway.

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