M002 · Kinematics / measurement

Free Fall & Measuring Gravity

Can a noisy camera and a coarse photogate timer measure gravity accurately? Precisely, yes, but not necessarily accurately. Fitting the drop height, initial speed and gravity together keeps the random error small. The timer's coarse ticks and air resistance can still push the result off in one direction, and more data will not fix that.

A calibrated drop tower, a movable photogate and a stroboscopic camera all watch the same fall. Change the planet, release height and initial velocity, fit the noisy camera positions, and use the residuals to tell precision from accuracy.

Exact inspection time $0\,\mathrm{s}$

Downward weight magnitude $0\,\mathrm{N}$

Signed upward air force $0\,\mathrm{N}$

Reference flag exit time $\text{Waiting}$

Amber ruler span $0\,\mathrm{m}$

Amber tick interval $0\,\mathrm{m}$

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 $0$

Record a fall and fit a model to it

Measure first, then test the model

Applied forcesBlue arrows show weight; violet arrows show the air or rod force. Force and velocity arrows each have their own scale, fixed for the whole trial. An instantaneous impact is not drawn as a finite force arrow.Metric rulerThe amber ruler uses the same length scale as the motion. Body sizes, frame thickness and the camera hardware are only schematic.Solved reference path so farThe green path is the solved physical reference; camera observations are recorded separately below.Vertical velocityThe teal apparatus arrow uses a fixed speed scale.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 the gold photogate. Every setting, including the time you inspect, also has a keyboard slider. Capture a new drop draws new noise; the physics stays the same.

Local gravity is uniform, the test mass is a 1 kg point and the ground is flat. Optional linear air resistance is there to make the constant-acceleration fit wrong on purpose. Camera noise is independent Gaussian noise in position, and clock error is modeled as quantization. The uncertainty shown is one standard error that assumes only the positions are noisy; it does not cover every source of error. The gate times a 10 cm flag. Its height in the 3D view is to scale, while the 2D flag is enlarged so it is easy to see. The instrument is simulated, not a real measurement.

Physics tutorial

A drop becomes an instrument

BackgroundWith a calibrated height and a clock, a falling body becomes a gravity experiment. The tower keeps three things apart: the actual trajectory, what the camera records, and the model fitted to those records. The photogate is a second instrument. It times how long a flag of finite length takes to pass, instead of recording a camera position.

Why it mattersThe free-fall equations in OpenStax University Physics, Volume 1, section 3.5 are exact for uniform gravity in a vacuum. A real measurement adds sampling, position noise and a finite clock resolution. Here you can see each of these effects, and you can switch on linear air resistance to make the model deliberately wrong.

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 sets a standard error for $\widehat g$ that assumes no other error; a coarse clock and air resistance can bias the result 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 is zero, but the gravitational acceleration still points down. The model stops the first time the body reaches the ground on the way down.

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

Time is centred and scaled before the least-squares fit, which avoids needless numerical trouble. The release height and velocity are unknowns fitted alongside gravity; they are not fixed 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, assuming the position errors are independent with equal variance and the observation times are exact. It leaves out clock quantization, calibration error and a wrong acceleration model. With fewer than four observations the residual variance cannot be estimated.

Photogate average speed and energy balance

\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, so the gate measures the average speed while the flag passes, not the speed at one point. If clock quantization rounds the transit time to zero, the gate reports the reading as unresolved. When air resistance is on, the energy balance includes the work it does.

Common difficulties

Small residuals do not guarantee 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 still look convincing while the gravity value is biased. A systematic pattern in the residuals 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 gives neighbouring frames the same timestamp and can round a short gate transit down to nothing. The standard error shown still assumes that only the positions are noisy.

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 comes close to the value you set, and the gate reports the flag’s average speed. In vacuum the energy balance error stays at round-off level.
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: Weaker gravity makes the experiment last longer. The three-parameter fit recovers the upward starting velocity instead of mistaking it for different gravity.
03
See the limits of noise and the clock
Choose Imperfect camera. Capture several drops, increase frame rate and change clock resolution separately.
What to observe: Each drop gets new noise, and the fitted value changes with it. Position scatter, timestamp quantization and unresolved photogate readings are separate 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: With drag, a constant-acceleration fit returns an effective acceleration. The bias comes from the wrong physical model, so reducing camera noise alone cannot remove it.