Fly on a Boat — Galilean Relativity

Source

Lecture: Lecture 0 — Prelude
Simulation: Fly on a Boat (Galilean relativity, non-relativistic)
Concept: Two inertial observers describe the same event with different trajectories but the same physics — velocities add by simple vector sum (Galilean).
Serves: Equivalence of inertial frames; frame-dependent vs frame-independent quantities; motivating question for the SR postulate to come.
Inquiry: Can Alice's straight drop and Galileo's parabola both be correct for the same fly? What quantity is the same in both frames, and what changes?
Misconception: "Galileo sees the fly drift backward / fall straight down" — ignoring that the fly already carries the ship's horizontal velocity at release.

Pedagogy principles applied

ID	Principle	How it shows up here
P1	Predict → observe → evaluate	Students sketch/predict the beach path BEFORE seeing the parabola, then read v_x and compare.
H2	Misconception encoded in a choice, not stated	The "straight down / drifts backward" errors are wrong choices with targeted rebuttals — not standing text.
A2/C1	Every number earns its place	Only two readouts: (x,y) and (v_x, v_y) in the ACTIVE frame — labelled with the frame.
A3	Formal tools off by default	Equations live in the ⚛︎ The Physics revision sheet, opened on demand; the sim is understandable purely from the picture.
E1	Frame-relevant display only	Switching frame CLEARS the trajectory and redraws it — you never see two inconsistent paths at once.
J2	Animation honesty	Fall time T = √(2h/g) and v_y(t) are identical in both frames; only v_x and the path differ.
P9	Bridge to next lecture	Closing question: would the same rule work for a light pulse? — motivates SR.

Fly on a Boat

Teach me

Speed

Side view — Cabin frame (Alice & Bella)

Fly

Trajectory

Ship / Cabin

Guided Inquiry

1 · A fly in a moving cabin

Let’s meet everyone. A ship glides along the coast at constant speed. Inside its closed cabin, Alice & Bella release a fly from rest at height h above the floor, while on the beach Galileo watches the same fly through the window.

Gravity g pulls the fly downward, and that’s all — nothing pushes it horizontally. Keep that in mind.

2 · Where does it land?

Here’s our first prediction. The fly is released from rest, high above the cabin floor, and the ship keeps gliding at the same speed the whole time. Where on the floor do you think it touches down?

3 · A faster ship

Now we’ll speed things up. The ship is re-set to glide at 9 m/s instead of 4, still perfectly steady, and Alice and Bella repeat the same drop. What do you think they measure now?

4 · The same fall from the beach

Time to step off the ship. Galileo watches that same drop from the beach, through the cabin window. What path do you think he sees the fly follow through the air?

5 · What both observers share

So Alice records a straight vertical drop, and Galileo records a forward-leaning parabola — one fly, one fall. What do you think — which measurement do the two of them agree on?

6 · Galilean relativity

Let’s give these ideas their names. The rule connecting the two descriptions is Galilean velocity addition: a velocity measured from the beach equals the velocity measured in the cabin plus the ship’s velocity. And the fact that every inertial observer gets a different path but the same physics is the equivalence of inertial frames.

Here’s one to wonder about: replace the fly with a pulse of light, and ask whether the beach observer measures c plus the ship’s speed. That’s where special relativity begins.

Listen

Frame

▾

Active observer

Controls

▾

Boat speed v_boat4.0 m/s

Release height h2.5 m

Gravity g9.8 m/s²

Fly · position (cabin)

▾

x0.00 m

y2.50 m

t0.00 s

Fly · velocity (cabin)

▾

v_x0.00 m/s

v_y0.00 m/s

Fall time T0.71 s

Cabin frame (Alice & Bella)

Beach frame (Galileo) — Galilean transform

Frame-independent (same in BOTH frames)

Galilean velocity addition

The Physics

Galilean relativity · what this sim shows, the physics behind it, and every control

At a glance

This sim releases a fly from rest inside a cabin on a ship that glides at constant velocity, and draws the fall as two observers would record it: Alice and Bella in the cabin, and Galileo on the beach. It introduces how two observers' descriptions of the same event are related.

What you'll learn

How the same fall is described by an observer in the cabin and by one on the beach.
How to convert positions and velocities from one frame to the other.
How to tell quantities that depend on the observer's frame from those that do not.
How the fall time depends on the release height, on gravity and on the boat speed.

A fly is released from rest at height h above the cabin floor. The ship moves at constant speed.

What's on screen

Fly (pink): released from rest at height h above the cabin floor.
Trajectory (blue): the path traced so far, in the frame being shown.
Velocity arrow (green, on the fly): the fly's velocity in the frame being shown.
Ship and cabin (amber), with Alice, Bella and a mast. Galileo stands on the beach. The arrow above the mast is the boat speed: amber, v_boat, in the Beach view; grey, v_beach = −v_boat, in the Cabin view.
Fly · position and Fly · velocity (sidebar): x, y, t, v_x, v_y and T, with the frame named in each panel title.

See itWatch the fall from the beach frame. Watch for: the pink fly falling from the cabin roof to its floor while the ship moves right, and the readouts counting time.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Frame	Whose description is drawn and read out: Alice and Bella in the cabin, or Galileo on the beach. Switching redraws the path so far in the new frame.	Cabin / Beach free exploration opens on Beach
Boat speed v_boat	The ship's constant speed relative to the beach. Moving the slider restarts the fall.	0–10 m/s opens at 4.0 m/s
Release height h	The height of the release point above the cabin floor. Moving the slider restarts the fall.	1–4 m opens at 2.5 m
Gravity g	The downward acceleration of the fly. Moving the slider restarts the fall.	2–20 m/s² opens at 9.8 m/s²
Fly · position	The fly's x (from the release point), its height y above the floor, and the time t since release, in the frame being shown.	panel
Fly · velocity	The fly's v_x and v_y in the frame being shown, and the fall time T from release to the floor.	panel
⏸ Pause / ▶ Play	Freezes the fall or resumes it. After landing, the fly rests on the floor briefly and is then released again.	top bar
Speed	Plays the fall faster or slower on screen. The physics and the readouts' values stay the same.	0.25×–4× opens at 0.25×
↻ Reset	Restores the controls to their opening values and restarts the mode you are in from its beginning.	top bar

Fixed: the fly starts at rest relative to the cabin, there is no air drag, and the ship's velocity never changes.

The fall in the cabin frame

The fly is released from rest relative to the cabin. Gravity pulls straight down, so no horizontal force acts on it. In the cabin frame, where the cabin and the fly both start at rest, the motion is a plain free fall from height h:

Cabin frame. x_c is measured from the release point, y_c upward from the floor.

Key idea. With no horizontal velocity and no horizontal force, x_c stays at 0 for the whole fall. The fly lands directly below the release point, whatever the boat speed.

See itSwitch to the Cabin frame. Watch for: the blue path is a vertical line, and x stays at 0.00 m while y falls to 0.

The same fall in the beach frame

Galileo uses beach coordinates. The two frames are related by the Galilean transformation (Lecture 6): time is the same for both observers, and the position along the ship is shifted by the distance the ship has moved.

Both x-axes point the way the ship moves and start at the release point; t = 0 at release.

Solving for x_b gives x_b = x_c + v_boat t = v_boat t:

Beach frame. The height and the vertical velocity are unchanged.

Horizontally the fly moves at the constant speed v_boat. Vertically it falls exactly as before.
Eliminating t gives y_b = h − g x_b² / (2 v_boat²): a parabola (for v_boat > 0).
The bend comes from the change of coordinates. Galileo sees no horizontal force either.

Both paths for the sim's opening values (v_boat = 4.0 m/s, h = 2.5 m, g = 9.8 m/s²).

See itSwitch to the Beach frame. Watch for: the blue path leans forward, and x grows steadily as the fly falls.

How velocities compare between frames

Differentiating x_c = x_b − v_boat t gives the velocity rule: the beach velocity is the cabin velocity plus the boat's.

u is the velocity measured on the beach, ũ the velocity measured in the cabin, and V = v_boat (Lecture 6).

Horizontal: v_x = 0 in the cabin frame and v_x = v_boat in the beach frame, for the whole fall.
Vertical: v_y = −gt in both frames, because the boat's velocity has no vertical part.
Speed: at the floor it is √(2gh) in the cabin frame and √(v_boat² + 2gh) in the beach frame (7.00 m/s and 8.06 m/s at the opening values).

See itWatch the velocity readouts in the Beach frame. Watch for: v_x stays at the boat speed while v_y runs steadily from 0 to −gT.

What sets the fall time

The vertical motion is the same free fall in both frames. Setting y = 0 gives the fall time, and x_b = v_boat t then gives the landing point:

The fall time is shared; only the landing coordinate depends on the frame.

v_boat h g

These sliders move the sim too.

T depends on h and g only. The boat speed does not appear in it.
T grows as √h: raising h from 1 m to 4 m doubles T (0.45 s to 0.90 s at g = 9.8 m/s²). T falls as 1/√g.

See itDrop the fly in the Beach frame at 4 m/s, then at 9 m/s. Watch for: the same fall time in both runs, with the landing x larger at 9 m/s.

Equivalent observers, different descriptions

Quantities the two observers agree on.

Frame-dependent: x, v_x, the shape of the path, and the speed. Each observer reports different values, and both are correct.
The same in both frames: t, y, v_y, the acceleration, the fall time T, and the event "the fly reaches the floor".
Because the acceleration agrees, F = ma reads the same in both frames, taking F and m to be frame-independent as Galilean physics does. A fall like this cannot reveal the ship's steady velocity to anyone inside the cabin.
Everything above rests on one assumption: time is absolute (t_c = t_b). Lecture 6 marks it as an assumption.

Key idea. The observers are equivalent, not identical: their numbers differ, their physics is the same, and neither is privileged.

See itSet the boat speed to 0 and look from the Beach frame. Watch for: the beach path is now a vertical drop, with x = 0.00 m and v_x = 0.00 m/s, as in the cabin.

Where it matters

Galileo's ship. In his Dialogue (1632), Galileo argued that flies, falling drops of water and jumps inside a closed cabin behave the same whether the ship is at rest or moving steadily. This sim is that argument.
A cruising airliner. A coin dropped from 2.5 m on a plane flying at about 250 m/s falls for 0.71 s and lands at your feet. In the ground frame it travels about 179 m in that time.
The Earth. The Earth moves around the Sun at about 29.8 km/s, some 21 km during a 0.71 s fall. Over so short a time that motion is very nearly steady, so its effect on a laboratory fall is negligible.

Go deeper: a pulse of light instead of a fly

Applying the velocity rule to a light pulse of speed c in the cabin would give the beach observer c + v_boat.
Lecture 7 shows that experiments (Michelson–Morley) disagree: Galilean velocity addition fails at speeds close to c. That is where special relativity begins.

Summary

In one line. One fall has two correct descriptions, related by x_c = x_b − v_boat t; the vertical motion and the fall time are the same in both.

Key points

In the cabin frame the fly has v_x = 0 and falls straight down. In the beach frame it has v_x = v_boat and follows a parabola.
Positions convert by x_c = x_b − v_boat t, and velocities by v_cabin = v_beach − v_boat.
T = √(2h/g) depends on h and g only.
Observers in relative motion at constant velocity are equivalent, not identical.

Common mix-ups

"Galileo sees the fly fall straight down, or drift backward." The fly keeps the ship's horizontal velocity, so from the beach its path leans forward.
"Something must push the fly forward to bend the path." No horizontal force acts in either frame. A force would be needed only to change v_x.
"One observer is really moving and the other is really at rest." Only the relative velocity matters here. Each observer's description is correct.

Things to try on your own

In the Beach frame, set v_boat = 10 m/s and h = 4 m. Use section 06 to predict where the fly lands, then compare with the x readout.
Find values of h and g for which the fall takes 0.50 s, and check the T readout.
Switch frames mid-fall and list which readouts change and which do not.