M002 · Kinematics / measurement

Free Fall & Measuring Gravity

A calibrated drop tower, movable photogate and stroboscopic camera share one trajectory. Change the planet, release height and initial velocity; fit noisy camera positions and watch the residuals reveal the difference between precision and accuracy.

Elapsed time $0\,\mathrm{s}$

Height $30\,\mathrm{m}$

Vertical velocity $0\,\mathrm{m\,s^{-1}}$

Gravity fit and standard error $\text{Collecting}$

Bias against truth $0\,\%$

Position residual RMS $0\,\mathrm{mm}$

Photogate speed $\text{Awaiting gate}$

Flag transit time $0\,\mathrm{ms}$

Energy balance defect $0\,\mathrm{J}$

Camera observations $0$

CALIBRATE / OBSERVE / FIT

Measure, then challenge the model

Continuous truth overlayCompare the fitted model with the solved trajectory.Release height

h_0

30.00\,\mathrm{m}

Initial vertical velocity

v_{y,0}

0.00\,\mathrm{m\,s^{-1}}

Local gravity

g

9.81\,\mathrm{m\,s^{-2}}

Camera frame rate

f_s

20.00\,\mathrm{Hz}

Position noise standard deviation

\sigma_y

0.01\,\mathrm{m}

Clock resolution

\Delta t_{\mathrm{clock}}

0.00\,\mathrm{s}

Photogate height fraction

h_{\mathrm{gate}}/h_0

0.50\,

Linear air damping

b/m

0.00\,\mathrm{s^{-1}}

Inspect flight

t/t_{\mathrm{impact}}

0.00\,\%

Drag the release ring or gold gate. All settings and time inspection also work with keyboard sliders. Capture a new drop changes the noise realization without changing the physics.

Uniform local gravity, point particle and a flat ground. Optional linear air resistance deliberately breaks the constant-acceleration fit. Camera noise is independent Gaussian position noise; clock errors are quantization. The displayed uncertainty is one standard error conditional on the position-only model, not a total error budget. The gate times a 10 cm flag, visually enlarged for readability; this is a synthetic instrument, not a real measurement.

Physics tutorial

A drop becomes an instrument

BackgroundA calibrated height and clock turn vertical motion into a gravity experiment. The tower separates the physical trajectory from camera observations and from the model fitted to those observations. A photogate is a second instrument: it measures the transit of a finite flag, rather than a camera position.

Why it mattersThe free-fall equations in OpenStax University Physics, Volume 1, section 3.5 are exact within a uniform-gravity vacuum model. Real measurement introduces sampling, position noise and clock resolution. This workbench makes those effects visible and lets you deliberately add a model mismatch with linear air resistance.

Start with the essentials

Focus question: Can you recover local gravity without pretending the initial position and velocity are known exactly?
One-sentence intuition: Fit the initial height, initial velocity and gravity together. Independent position noise gives a conditional standard error for $\widehat g$ ; quantized time and air resistance can create bias even when that standard error is small.

Core mathematical model

The vacuum reference

y(t)=h_0+v_{y,0}t-\frac12gt^2,\qquad v_y(t)=v_{y,0}-gt

Upward is positive. At the apex the velocity vanishes, but the gravitational acceleration remains downward. The model stops at its first descending ground crossing.

Fit three parameters

\min_{\beta_0,\beta_1,\beta_2}\sum_i\left[y_i-(\beta_0+\beta_1t_i+\beta_2t_i^2)\right]^2,\qquad\widehat g=-2\beta_2

Centering and scaling time protects the least-squares solve from unnecessary numerical conditioning problems. The release height and velocity are fitted nuisance parameters, not secretly held at their true values.

Conditional uncertainty

\operatorname{SE}(\widehat g)=2\sqrt{\widehat\sigma^2[(X^{\mathsf T}X)^{-1}]_{33}},\qquad\widehat\sigma^2=\frac{\sum_i r_i^2}{N-3}

This is one standard error under independent equal-variance position errors and exact observation times. It does not include clock quantization, calibration error or an incorrect acceleration model. Fewer than four observations cannot estimate the residual variance.

Finite photogate and energy audit

\overline v_{\mathrm{gate}}=\frac{\ell}{t_{\mathrm{exit}}-t_{\mathrm{entry}}},\qquad E(t)-E(0)-W_{\mathrm{air}}(t)=0

The flag is 10 cm long; a finite transit reports an average speed, not an exact point velocity. Clock quantization can round the transit to zero, in which case the instrument reports an unresolved reading. The energy ledger includes optional air work.

Common difficulties

A small residual is not a certificate of accuracy

Typical misconceptionA smooth parabola and a small standard error mean the reported gravity must be correct.

Better mental modelTurn on air damping. The fitted parabola can look convincing while gravity is biased; systematic residual structure is evidence against the constant-acceleration model.

More frames do not repair the clock

Typical misconceptionIncreasing frame rate must improve every measurement.

Better mental modelA coarse clock assigns repeated timestamps to adjacent frames and can erase short gate transit times. The displayed standard error remains conditional on a position-only noise model.

The upward launch is still free fall

Typical misconceptionGravity starts acting only after the ball reaches its highest point.

Better mental modelThe same downward acceleration applies throughout ascent and descent in vacuum. The velocity changes sign; acceleration does not.

Run the experiment

01
Calibrate on Earth
Choose Earth tower, set position noise and clock resolution to zero, then capture a drop. Move the gate and compare its speed with the velocity readout at the crossing.
What to observe: The fitted gravity approaches the chosen truth; the finite gate reports the flag’s average speed. The vacuum energy defect stays near roundoff.
02
Change the planet and the initial motion
Compare Moon tower with Earth tower, then choose Upward launch and scrub through the apex.
What to observe: Lower gravity lengthens the experiment; an upward initial velocity is recovered by the three-parameter fit instead of being mistaken for different gravity.
03
Expose noise and clock limits
Choose Imperfect camera. Capture several drops, increase frame rate and change clock resolution separately.
What to observe: Noise realizations change the fitted value. Position scatter, timestamp quantization and unresolved photogates are distinct sources of error.
04
Break the fitted model
Add linear air damping and inspect the fitted gravity and residual shape, then restore vacuum.
What to observe: A constant-acceleration fit estimates an effective acceleration under drag. The bias is physical model error; reducing camera noise alone cannot remove it.