Projectile Motion — Independent x and y

Source

Simulation: Projectile Motion — Independent x and y (live launch + vertical-twin race + range-vs-angle function plot, with a runnable counterfactual "coupled" model).
Concept: Horizontal and vertical projectile motions are independent: x(t) = v₀cosθ·t at constant velocity; y(t) = v₀sinθ·t − ½gt² under gravity alone (SI units, g = 9.81 m/s²).
Serves: Component decomposition of 2-D motion; R = v₀²sin2θ/g peaking at 45° (R_max = v₀²/g); complementary angles (30°/60°) give equal range. Kernel-verified anchors: 25 m/s @ 40° → R = 62.74 m, T = 3.28 s, H = 13.16 m; 30° vs 60° → 55.17 m = 55.17 m; 45° → 63.71 m.
Inquiry: Does the tie-line between the projectile and its vertical twin ever tilt as v₀ grows? Which velocity arrow changes in flight? Which angle maximizes range, and why do θ and 90°−θ land at the same spot but at different times?
Misconception: "Horizontal and vertical motions influence each other — a faster horizontal throw falls more slowly." Encoded as prediction choice ⓐ and as the runnable ⚠ Coupled model (a_y = −g·sinθ), falsified by contrast with the ✓ Independent model.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept, one sentence	Everything serves one claim: x and y motions of a projectile are independent.
A3/A4	Formal tools off by default	All equations sit behind the ⚛︎ The Physics button (a pop-up revision sheet, closed on load); the sim is fully playable pre-formalism.
C1/C2	Readout discipline	Two result badges only (t and R), each used by the predict/explore prompts; v components, T, H live in the collapsed Data panel. Speed sits beside The Physics in the top bar like the other shell sims.
D6/D7	Stepwise disclosure, never gating	Twin ball, 90°−θ comparison, range plot and ⚠ mode are additive reveals via always-enabled controls; nothing is locked behind steps or predictions.
E1/J2/J5	Animation honesty	Exact closed-form kinematics (no integrator); repeat passes are fresh launches, never reversed; naive θ=0 "glide" clips with a → ∞ label; range-plot axes are fixed; the scene scale only grows automatically (Fit/Clear re-frame on demand).
G4b	Explicit reveal control	The prediction's answer (the twin race) appears only when the student flips ⇅ Twin ball — one always-available click.
H1/H2	Structural misconception	"Faster horizontal throw falls more slowly" is prediction choice ⓐ and the exact ⚠ Coupled model (a_y = −g sinθ); the rebuttal appears only if chosen. No standing myth card.
—	Prediction-first, non-spoiling	Twin is off on load so the opening animation never reveals the tie; R shows "—" until the first landing; prediction is optional and never blocks play.
—	Counterfactual contrast	Both models implemented exactly; in ⚠ mode the thin ✓-real arc is overlaid for direct comparison.
—	Machine-checkable physics	window.__audit.at() returns R, T, H from the same functions the display uses (g = 9.81 m/s²).

Projectile Motion

Teach me Hide Text

Speed

A projectile

Guided Inquiry

1 · One launch

Let’s meet our ball. Ball A leaves the ground at v₀ = 25 m/s and θ = 40° above the horizontal. You’ll see its velocity drawn in two pieces, a horizontal component and a vertical component, while gravity acts straight down with g = 9.81 m/s².

2 · The twin race

Now for a race. At the same instant, a second ball is thrown straight up with the same vertical speed v₀sinθ and no horizontal velocity at all. Which ball do you think reaches the ground first?

3 · Two angles, one speed

Let’s play with the angle instead. Two balls are launched at the same speed, one at 30° and one at 60°. How far from the launch point do you expect each one to land?

4 · The longest shot

Now take that one step further. 30° and 60° share a range, and so does every other pair of angles adding to 90°. So which single launch angle do you think sends the ball farthest?

5 · Where, and when

Time to watch it happen. Both balls are now launched at 25 m/s, one at 30° and one at 60°. Compare where each one lands — and then compare when each one lands. Notice anything?

6 · Doubling the launch speed

One last question. The launch angle is held fixed, and the launch speed v₀ is doubled. By what factor do you think the range changes?

Listen

Controls

▾

Launch speed v₀25.0 m/s

550 m/s

Angle θ40°

0°90°

Actions

Compare

Show axes & grid

Readouts

▾

Time since launch

— s

Range last ✓ landing

— m

📈 Range vs angle

▾

📊 Data (closed form)

▾

vₓ₀ = v₀cosθ	—
v_y₀ = v₀sinθ	—
Flight time T	—
Max height H	—
Range R	—
⚠ coupled T	—
⚠ coupled R	—

Exact values for the current sliders (✓ model): T = 2v₀sinθ/g, H = (v₀sinθ)²/2g, R = v₀²sin2θ/g, with g = 9.81 m/s².

Horizontal: no horizontal force ⇒ constant velocity. Gravity never touches x.

Vertical: free fall. vₓ does not appear anywhere in y(t) — that absence is the independence.

Hang time and peak height come from the vertical component alone — the twin ball shares both exactly.

Range: constant vₓ times the vertical clock. Peaks at θ = 45° (sin2θ = 1 ⇒ R_max = v₀²/g).

Complementary angles land at the same spot — 30° and 60° share R (55.17 m at v₀ = 25 m/s) but not T.

⚠ The falsified "coupled" model behind the myth: hang time independent of angle, flat throws never land. Runnable via the ⚠ button — and refuted by every real fall.

The Physics

Projectile motion · what this sim shows, the physics behind it, and every control

At a glance

This sim launches a ball from flat ground at the speed and angle you choose, draws its arc, and splits its velocity into a horizontal and a vertical part. Extra balls let you race launches against each other, which is how you find out what the arc really depends on.

What you'll learn

What the horizontal and vertical parts of the velocity each do during the flight.
What sets how long a ball stays in the air.
How the range depends on the launch angle.
How the range depends on the launch speed.

Ball A leaves the ground at speed v₀ and angle θ. Its launch velocity splits into a horizontal part (amber) and a vertical part (blue).

What's on screen

Ball A (amber ring): your launch. Its arrows show vₓ (amber) and v_y (blue).
Launcher: the v₀ arrow at angle θ, split into its two parts.
Ball B (cyan): the twin, from ⇅ Twin ball.
Ball M (violet): the ball at 90° − θ, from ⧉ 90°−θ.
Previous trials: faint arcs of earlier launches.
Landing flags: each ball's flight time and range.

See itLaunch ball A at the current settings. Watch for: the landing flag, which shows the flight time and the range.

The controls

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

Control	What it changes	Range
Launch speed v₀	How fast the ball leaves the launcher.	5–50 m/s opens at 25 m/s
Angle θ	The launch angle above the horizontal.	0°–90° opens at 40°
🚀 Launch	Fires a new run at the current settings. The previous arc stays as a faint trial (up to four).	button
↻ Auto	When on, letting go of a slider or switching a comparison ball fires a new run straight away. When off, changes wait for 🚀 Launch.	on / off opens off
🧹 Clear	Removes the previous trials and refits the view.	button
⇅ Twin ball	Adds ball B, thrown straight up at the same instant with the same vertical speed v₀ sin θ and no horizontal speed.	on / off
⧉ 90°−θ	Adds ball M, launched at the same speed and at angle 90° − θ.	on / off
Show axes & grid	Draws metre axes and a grid on the scene.	on / off
Readouts	Time since launch, and the range of ball A's last landing.	panel
📈 Range vs angle	Plots range against launch angle for the current v₀, with a marker at your θ.	panel
📊 Data	Exact vₓ, v_y, flight time, peak height and range for the current sliders.	panel
▶ Play / ⏸ Pause	Freezes or resumes the flight. After a landing, ▶ Play launches again.	top bar
Speed	Plays the flight faster or slower on screen. The physics is unchanged.	0.25×–4×
↻ Reset	Restores the defaults and restarts the mode you are in.	top bar

Fixed: g = 9.81 m/s², no air drag, and every ball is launched from and lands on the same flat ground.

Two motions share one clock

After launch the only force is gravity, and it points straight down. So the ball has no horizontal acceleration, and a constant downward acceleration g.

The opening launch, 25 m/s at 40°. The horizontal velocity never changes; the vertical one shrinks, vanishes at the top, then grows downward.

Horizontal: no force, so constant velocity.

Vertical: free fall. vₓ appears nowhere in y(t).

The two motions share only the time t. Neither one affects the other.
x grows steadily with t while y is quadratic in t, so the path is a parabola.
At the top only v_y is zero. The ball still moves sideways at v₀ cos θ, and its acceleration is still g, downward.

See itLaunch ball A with its velocity arrows. Watch for: the amber arrow keeps the same length all flight, while the blue one shrinks to nothing at the top and then grows pointing down.

What sets the flight time

The flight time and the peak height come from the vertical motion alone:

Set by v₀ sin θ and g only.

Twin ball B starts with the same vertical speed v₀ sin θ as A, but no horizontal speed. Its y(t) is the same as A's, term by term.
So the two balls are at the same height at every instant, and land together. At 25 m/s and 40° both land after 3.28 s.
A's horizontal speed only carries it sideways. It does not keep A up any longer.

The twin race at the opening launch, with two same-height moments marked.

See itRace ball A against twin B. Watch for: the dashed line between the balls staying level ("same height"), and both landing flags showing the same time.

Range is drift times flight time

The ball drifts sideways at a steady v₀ cos θ for the whole flight time T, so the range is their product:

Valid for launch and landing at the same height.

v₀ θ

These sliders move the sim too.

Launch speed feeds both factors: double v₀ and the drift doubles and so does T. The range grows four times (R ∝ v₀²).
At 40°: 25 m/s gives 62.74 m, and 50 m/s gives 250.97 m.

See itLaunch at 40° with 25 m/s, then with 50 m/s. Watch for: the second ball lands four times as far, and the card shows both ranges side by side.

How the angle sets the range

The angle enters the range only through sin 2θ, and sin 2θ takes the same value for θ and 90° − θ:

At 25 m/s, 30° and 60° both land 55.17 m away. The steeper ball stays up longer (4.41 s against 2.55 s) but drifts more slowly, and the two effects cancel.
45° pairs with itself, and there sin 2θ = 1 is largest: R_max = v₀²/g, which is 63.71 m at 25 m/s.

Three angles at 25 m/s.

See itLaunch at 30° and 60° together. Watch for: both flags at the same range, with different flight times.

See itLaunch at 45° with 📈 Range vs angle open. Watch for: the amber marker sits at the top of the green curve.

Where it matters

Throwing events. Shot-putters release well below 45°, typically in the mid-30s. The shot leaves from about 2 m above the ground, and athletes can throw faster at flatter angles. Both break the sim's assumptions.
Air drag. For a fast, light ball such as a hit baseball, drag shortens the flight a lot and pulls the best angle below 45°. The sim leaves drag out so the two motions stay separate.
On the Moon, g = 1.62 m/s². Since R ∝ 1/g, the same 25 m/s throw at 45° would land about 386 m away, six times farther, after about 22 s in the air.

Go deeper: the shape of the path

Take t = x / (v₀ cos θ) from the horizontal motion and put it into y(t). The time drops out: y = x tan θ − g x² / (2 v₀² cos² θ).
This is a downward parabola in x. It starts with slope tan θ, peaks halfway along at x = R/2, and lands with slope −tan θ.

Summary

In one line. A projectile drifts sideways at constant speed while it falls freely, and its range is that drift speed times the flight time.

Key points

aₓ = 0 and a_y = −g, so vₓ = v₀ cos θ never changes.
Flight time T = 2v₀ sin θ / g depends only on the vertical launch speed.
R = v₀² sin 2θ / g: angles θ and 90° − θ share a range, and 45° gives the most.
Doubling v₀ doubles T and quadruples R and H.

Common mix-ups

"A faster sideways throw stays up longer." vₓ does not appear in y(t), so the twin lands at the same moment.
"At the top the ball is at rest." Only v_y is zero there. vₓ is unchanged, and the acceleration is still g.
"Double the speed, double the range." Drift and flight time both double, so the range quadruples.

Things to try on your own

Launch at 90° with the twin on, and compare the two balls.
Launch at 30° with 20 m/s, then 40 m/s, and compare the ranges.
At 25 m/s, find the two angles that both land 31.86 m away.