For high-school explorers · physics you can fly

How does a drone hang in the air?

A quadrotor is just four spinning fans on a cross, and yet it can hover dead still, dart sideways, and spin on the spot. The secret isn't magic; it's physics you can learn from scratch. We start at the very beginning, pushes and pulls, Newton's laws, twists (torque), build up how things move and why, and end by flying a real quadrotor: hover, lean to move, and spin to turn. Every idea comes with an animation, a hands-on slider, a worked number, and a friendly equation. No physics required to start.

🎮 hands-on labs 🎞️ animated 🔢 worked examples 🧮 friendly math 🚁 build to real flight

Start the journey Jump to the labs

How to read this

The toolbox → how things move → real flight.

Guided lesson: one idea at a time, plain words first, a picture right beside it.
A friendly equation: every formula gets a plain-English reading and a symbol key.
A worked example: the idea run with real numbers, step by step.
Try it yourself: drag the sliders in the play labs and watch the physics respond.
Puzzles: a couple of questions per idea, with hints and full solutions.

The deck · 13 ideas, in order

Pick a concept.

Read top-to-bottom like a story: first the physics toolbox (forces, Newton, torque, rigid bodies), then how things move (kinematics, dynamics, spinning), then a real quadrotor (body vs world frames, anatomy, hover, roll & pitch, yaw, the full control loop).

🗺️ Your flight path. Thirteen ideas in three stages: a little drone flies the route, and the glowing stop is where it is now. Tap any stop to jump to that idea.

1 · The toolbox2 · How things move3 · Real flight

✦ The big one

A net force makes mass accelerate

\[ \vec F_{\text{net}} = m\,\vec a \]

Newton's second law is the engine of the whole module. Add up the force arrows on the drone (thrust up, gravity down, a little drag), divide by the mass, and you get the acceleration, which tells you exactly how its motion changes next. Rearranged as a = F/m, it's the single rule a flight computer leans on.

✦ The spinning twin

A net torque makes a body spin up

\[ \tau = I\,\alpha \]

Rotation copies straight-line motion symbol-for-symbol: force becomes torque, mass becomes moment of inertia, acceleration becomes angular acceleration. Make one side of the drone push harder and the leftover torque tips it. That's how it rolls, pitches, and yaws. Learn one equation, get the other free.

✦ The flying trick

Lean a little, fly sideways

\[ a_{\text{sideways}} = g\,\tan\theta \]

A drone's thrust only points along its own "up", so to go sideways it tilts. Hold your height while leaning by angle θ and the sideways acceleration comes out beautifully simple, just gravity times the tangent of the lean. No mass, no thrust in sight. It's why every drone leans into the direction it flies.

Play · watch the math move

Nine labs that turn the knobs into math.

Drag a slider and three things move together: the picture, the live equation (with the real numbers dropped in), and a plain-English why it does that to the drone. The colours in the equations match the arrows on screen, so you can see exactly which number is which.

thrustweightnet forceholds-up partsideways part

Capstone project · write code, press Run, watch it fly

Simulate a quadrotor in Python.

This pulls the whole module together. Below is a tiny flight computer you program in Python-style code. Press Run and a real simulator flies exactly what you wrote, then tells you if the mission succeeded and, if not, what went wrong. Five missions climb from sequencing a flight plan, to tuning the controls, to writing the control law yourself.

The state (kinematics)

A quadrotor's state is where it is and which way it points: position \((x, z)\) with velocities \((v_x, v_z)\), plus a tilt (pitch \(\theta\)) and a heading (yaw \(\psi\)) with their rates. Kinematics just tracks how those numbers change: velocity is the rate of position, acceleration the rate of velocity. Position and heading live in the world frame; thrust lives in the body frame, so the sim rotates between them every step.

The dynamics (why it moves)

Forces and torques set the accelerations, through \(\vec F = m\vec a\) and \(\tau = I\alpha\). The four rotor thrusts add up to a total lift \(T\) and create the steering torques. Tilt the body and that single thrust splits into an up part and a sideways part, which is how a quad flies anywhere.

The equations of motion

\[ m\ddot z = T\cos\theta - mg \]

\[ m\ddot x = T\sin\theta \]

\[ I\ddot\theta = \tau_\theta, \qquad I_z\ddot\psi = \tau_\psi \]

The simulator steps these forward every \(0.02\) s. Your code decides the thrust and the torques (directly, or through the commands and gains it hands the autopilot).

Where to go next

Books & sources.

The physics here is standard high-school / first-year mechanics; the quadrotor part follows the modern robotics references.

Halliday, Resnick & Walker, Fundamentals of Physics, forces, Newton's laws, torque, rotation (the whole toolbox).
Hibbeler, Engineering Mechanics: Statics & Dynamics, rigid bodies, moment of inertia, and motion.
Mahony, Kumar & Corke, “Multirotor Aerial Vehicles,” IEEE Robotics & Automation Magazine (2012), the standard quadrotor primer.
Beard & McLain, Small Unmanned Aircraft: Theory and Practice, Princeton (2012).
Bouabdallah, “Design and control of quadrotors,” EPFL PhD thesis (2007).

Related rooms

Where this leads.

Reinforcement Learning & Control Probability for Robotics World Models All missions