---
title: "Terminal Fall & Drag Laws Interactive Lab | Sandbox Physics"
description: "Can two falling masses reveal the drag law? Yes."
canonical_url: "https://sandboxphysics.com/labs/terminal-fall/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/terminal-fall/"
last_modified: "2026-10-27"
---

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.

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

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

Physics tutorial

## Terminal Fall & Drag Laws

**Background** Gravity pulls down, and the resistance grows with speed. Each drag law gives terminal speed its own dependence on mass and its own way of approaching that speed.

**Why it matters** Drop two bodies with equal area and different masses, compare linear and quadratic drag and start a fall above terminal speed. Then try to infer the drag from a sampled speed record.

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 misconception** Once I see the terminal speed, I know the whole drag law.

**Better mental model** A 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Terminal speed is where the forces balance. A noisy record of the approach is evidence, but limited evidence.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

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)