Dodgeball — Galilean Velocity Addition

Source

Lecture: 6 — Galilean Relativity
Simulation: Simulation 1 — Dodgeball (guided inquiry)
Concept: The velocity of an object depends on the observer's reference frame; Galilean velocity addition v′ = v − V_frame transforms velocities between any two inertial frames, with none being privileged.
Serves: Elements: relative motion, relative speed, relative velocity. Outcomes: describe relative motion between inertial observers; apply Galilean transformations to positions and velocities; explain the principle of Galilean relativity.
Inquiry: Requiring students to select a frame before reading velocities makes frame-dependence unavoidable by design; the adjustable velocities of A and B let students explore Galilean velocity addition.
Misconception: "The ground frame is the natural one to use — its velocities are the real ones." Surfaced only via prediction option (c) and the friend-persona prompt.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Everything serves frame-dependence of velocity via Galilean addition — no forces or acceleration.
A2	No decoration	Court lines, arrows, readouts and the frame indicator all carry meaning.
A3/A4	Formal layer optional	The Physics sheet stays closed by default; the sim works fully from arrows + readouts alone.
B3	Symbol definitions	v_A, v_B, v_ball,A, V_frame, v′ defined in readout labels and The Physics sheet.
C1	Numbers earn place	Each velocity is read/predicted/compared; the animation-speed control is hidden.
C2	Frame-labelled quantities	Readouts state object, frame and units; the indicator gives the frame's own ground velocity.
D1	Primary visual	The top-down field is the largest element; controls are subordinate.
D3	Identifiable entities	A = blue, B = red, ball = orange, each labelled.
D5	No clipping at extremes	Arrows are capped/clamped; off-view objects show an edge marker; readouts wrap.
D6	Incremental complexity	Steps add comparison overlays and persona emphasis — never disabling anything.
D7	Playable from load	All sliders, frame tabs, Play and Predict work with zero clicks.
D8	Labels not narration	Canvas shows labels/arrows/indicator only; prose lives in panels.
E1	Faithful animation	Constant-velocity flight; ball resets to A only after reaching B or leaving the field.
E3	No extras	No sparkles/trails; only the meaningful velocity arrows update live.
F2	Conceptual prediction	Predict asks "same vs changes," not a numeric answer.
F6	Confirm-what-you-see	Step 1 invites noting the ball's speed before switching — a suggestion, not a gate.
G3	Frame switch = re-render	Arrows, scroll rate and every number change; the rest object goes still.
G4 / G4b / G4c	Frame behaviour	Active frame's object is at rest; one frame at a time; identity/order preserved.
G5	Live numbers in prompts	The prediction prompt reads the current v_ball,A slider value.
G7	Operational velocity	The Physics sheet shows v′ = Δx′/Δt measured in the chosen frame.
H1 / H2 / H3	Structural misconception	"Ground is special" appears only as option (c)'s reveal and the friend persona.
J1	Consistent notation	v with subscripts; v′ for transformed; V_frame for the frame's ground velocity.
J3	Correct model	v′ = v − V_frame; the catch event is frame-invariant in time.

Dodgeball

Teach me

Speed

A's rest frame

Frame velocity: +2.0 m/s → (rel. ground)

→ = positive (rightward) · ← = negative

Velocities in A's rest frame→ = positive (rightward)

Aat rest

B+3.0 m/s

ball+8.0 m/s

Guided Inquiry

1 · A throw across the court

Let’s start with a game of dodgeball. A and B move along a straight court, each with a velocity measured against the ground. A throws the ball forward at v_ball,A = 8.0 m/s — a speed measured relative to A, not to the ground. Press Play and watch one throw go.

2 · Does the thrower's own motion count?

Here’s the first question. A is running forward at the instant of the throw. Measured against the ground, how fast do you think the ball travels?

Try giving A more ground velocity: you’ll see B’s reading in A’s frame drop, and the ball catch B sooner.

3 · The same ball, measured by B

Now let’s change who’s measuring. In A’s rest frame the ball travels at 8.0 m/s. B measures that same ball from her own rest frame. What do you think B gets?

Go ahead and switch between the three frames — notice that every velocity changes at once.

4 · How fast the gap closes

Now push that idea further. Every frame gives the ball a different velocity, and B a different velocity too. So what do you think happens to the rate at which the gap between ball and B closes?

Try it yourself: in each frame, subtract B’s velocity from the ball’s — the answer never changes.

5 · Galilean relativity, and where it ends

Let’s pull it all together. Every velocity here follows v′ = v − V_frame. Frames disagree about each object’s velocity but agree about the differences between them, so all three descriptions of this throw share one catch.

Here’s the twist to take with you: Galilean addition has no ceiling. Relativity replaces it with w = (u + v)/(1 + uv/c²), which never reaches c. Under that rule the closing rate v_ball − v_B does change from frame to frame, but the ball’s velocity relative to B, the reading B takes in her own rest frame, still comes out the same whichever frame you work it out from. This simulation models only the Galilean rule.

v′ = velocity in the active frame · V_frame = that frame's own ground velocity

Listen

Controls

▾

Reference frame

A v (ground) +2.0 m/s

B v (ground) +5.0 m/s

Ball v (rel A) 8.0 m/s

Galilean velocity transformation (v′ ≡ velocity in active frame; V_frame ≡ frame's ground velocity)

Active frame's velocity relative to the ground

Person A

Person B

Ball (ground velocity = v_A + v_ball,A)

Velocity is a per-frame measurement

The Physics

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

At a glance

Two players run along a court while A throws a ball at B, and the sim lets you read every velocity from A's, B's or the ball's own rest frame. It is the first step towards relativity: before asking what changes near the speed of light, you need to know exactly what a velocity means at everyday speeds.

What you'll learn

How the ball's velocity over the ground is related to A's motion and to the throw.
What each rest frame reports for A, B and the ball, and why.
Which quantities depend on the choice of frame, and which do not.

A throws at B. v_A and v_B are measured against the ground, v_ball,A against A. Positive is rightward.

What's on screen

A (blue): the thrower.
B (red): the target.
Ball (orange): starts at A. The throw restarts when the ball reaches B, or when the ball or B leaves the court.
Velocity arrows and court markings: the arrows show each object's velocity in the active frame. The floor ticks stay still or slide, depending on that frame.
Frame badge (top right of the court): the active frame, and its own velocity relative to the ground.
Velocities strip (under the court): the numbers for A, B and the ball in the active frame.

See itStart one throw in A's rest frame. Watch for: the ball leaving A, reaching B, and the throw starting over.

The controls

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

Control	What it changes	Range
Reference frame	Whose rest frame the numbers are read in: A's, B's or the ball's. The velocities, arrows and floor markings are redrawn for that frame. It does not restart the throw.	3 buttons opens in A's frame
A v (ground)	A's velocity relative to the ground. Moving any velocity slider restarts the throw.	−8 to +8 m/s opens at +2.0
B v (ground)	B's velocity relative to the ground.	−8 to +8 m/s opens at +5.0
Ball v (rel A)	The throw speed, measured from A.	1 to 15 m/s opens at 8.0
Frame badge	Names the active frame and gives that frame's own velocity relative to the ground.	on the court
Velocities strip	A, B and the ball, each in the active frame.	under the court
▶ Play / ⏸ Pause	Runs or freezes the throw.	top bar
Speed	Plays the throw faster or slower on screen. No readout changes.	0.25×–4× opens at 1×
↻ Reset	Puts the three sliders and the frame back to their opening values and restarts the throw.	top bar

Fixed: the court is 44 m long, and each throw starts with A at 10 m and B at 30 m. All velocities are constant.

The ball carries the thrower's motion

The sliders A v and B v are set against the ground. The throw speed is set against A, so it is not yet a ground velocity. The ball leaves A already moving with A, so over the ground the two velocities add:

Velocities over the ground. Positive is rightward.

At the opening settings: 2.0 + 8.0 = 10.0 m/s over the ground, although A's throw alone is 8.0 m/s.
No button selects the ground as a frame, but the A v and B v sliders are set in it.
Every rest frame the sim offers reads the ball from this one ground value (next section).

See itSwitch to the ball's rest frame. Watch for: the Frame velocity badge reading the ball's ground velocity, +10.0 m/s at the opening settings.

Each frame subtracts its own velocity

Two observers share one clock, and their rulers are offset by the motion between them (Lecture 6, with V the frame's velocity):

Differentiating the position once gives the rule for every velocity. The lecture writes it ũ = u − V; the sim writes it as:

V_frame is the ground velocity of whatever the frame rides: v_A, v_B, or v_A + v_ball,A for the ball.
Every velocity loses the same V_frame, sign included, so the object the frame rides reads zero.
The court markings are not exempt: they read −V_frame.

One throw at the opening settings, read in A's frame and in B's frame (m/s).

See itSwitch to B's rest frame. Watch for: B at rest, A and the ball re-read, and the floor markings sliding backwards. At the opening settings A reads −3.0 and the ball +5.0 m/s.

Differences are the same in every frame

Subtract two velocities read in the same frame, and V_frame cancels:

The rate at which the ball closes on B is v_ball − v_B = 10.0 − 5.0 = 5.0 m/s in every frame, though the ball and B each read differently.
The same holds for any pair: B moves away from A at v_B − v_A = 3.0 m/s in every frame.
No frame has the "true" closing rate. Each one just reads it off a different pair of numbers.

The same throw in each frame. The whole axis slides; the teal gap, the closing rate, does not.

See itSwitch to the ball's rest frame. Watch for: the ball held still while B reads −5.0 m/s at the opening settings, the same 5.0 m/s gap rate.

All frames agree on the catch

The Galilean rule keeps one shared clock, t̃ = t. At the opening settings the ball starts 20 m behind B and closes at 5.0 m/s, so every observer clocks the catch at 20 ÷ 5.0 = 4.0 s.

v_A v_B v_ball,A

These controls move the sim too.

The acceleration agrees as well. V is constant, so it vanishes when ũ is differentiated, and Newton's second law reads the same in every frame:

See itRun one throw in each of the three frames, one after another. Watch for: three different sets of velocities in the strip, and the same catch time on every run.

Where the rule stops

Plain addition and subtraction fail for light. If Alice clocks a light pulse at c, Bob moving along with it at V should clock c − V, but every measurement gives c. Lecture 7 resolves this. Relativity replaces the addition rule, with c = 3.00 × 10⁸ m/s:

When uv is much smaller than c², the denominator is 1 and the rule is u + v.
Put u = c and the rule gives w = c for any v.
In this sim the correction is about 2 parts in 10¹⁶ at the opening settings, and about 1 part in 10¹⁵ at the fastest settings (A v = 8, ball v = 15 m/s). No stopwatch could see that, so the sim models only the Galilean rule.

Where it matters

Airspeed and ground speed. A plane at 200 km/h next to a train moving the same way at 100 km/h is seen from the train at 100 km/h (the lecture's example). With a wind along the flight path, ground speed is airspeed plus wind, by the same arithmetic.
Collisions. Two cars in the same lane at 20 m/s and 15 m/s meet at a closing speed of 5 m/s. The severity of the impact depends on that difference, not on either speed over the road.
Galileo's ship. Because ã = a, an experiment inside a smoothly moving cabin gives the same results as on shore. This is the principle of relativity, which Einstein kept and extended.

Go deeper: the relativistic version of the subtraction

Beyond this sim, the rule u′ = u − V becomes u′ = (u − V)/(1 − uV/c²). For small speeds it is the sim's rule, and for u = c it gives u′ = c in every frame.

Summary

In one line. Changing frame subtracts the same V_frame from every velocity, so velocities depend on the frame while differences, time and acceleration do not.

Key points

The ball's ground velocity is v_A + v_ball,A.
In any rest frame v′ = v − V_frame, and the court markings read −V_frame.
The closing rate v_ball − v_B and the catch time are the same in every frame.
The rule is exact for everyday speeds and wrong for light.

Common mix-ups

"The ground frame gives the ball's real speed." The ground is one more moving object. In every frame the court markings read −V_frame.
"Switching frames changes the motion." It changes only the description. A, B and the ball stay where they were, and the catch still happens.
"If frames disagree on velocities, they disagree on everything." They agree on differences of velocity, on time, and on acceleration.

Things to try on your own

Raise A v and B v by 3 m/s each and stay in A's frame. Compare A, B and the ball in the strip with before, then look at the floor markings.
Set A v = 2, ball v = 6 and B v = 8. Work out the closing rate before you press Play, then check what the sim does.