---
title: "Free Fall & Measuring Gravity Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/free-fall/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/free-fall/"
last_modified: "2026-10-22"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

**Background** With 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 matters** The 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 misconception** A smooth parabola and a small standard error mean the reported gravity must be correct.

**Better mental model** Turn 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 misconception** Increasing frame rate must improve every measurement.

**Better mental model** A 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 misconception** Gravity starts acting only after the ball reaches its highest point.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Every measurement has three parts: the apparatus, the way it samples, and the model used to interpret the samples. Look at all three before you trust the final number.

Starter path · Step 1 of 6

## From a falling ball to an orbit

[Next step → **Projectile Flight Designer** Launch in vacuum first, then switch on drag and compare the range and apex with the vacuum reference. Compare · 16 min](https://sandboxphysics.com/labs/projectile-flight/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)