At what angle is range maximum (flat ground)?

R = v₀² sin(2θ)/g is maximum when sin(2θ)=1 → θ=45°.

From x-t data, determine the launch speed if angle=30°.

V_x = v₀cos(30°); slope of x vs t gives v_x; v₀ = v_x/cos(30°).

On the Moon (g = 1.6 m/s²), how does the range of the same launch compare with Earth's?

R = v₀²sin(2θ)/g scales as 1/g: 9.81/1.6 ≈ 6.1× longer. Verify with the Moon Shot preset.

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Pro 🔒~25 min

Projectile Data Lab

Collect and analyze real projectile motion data

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How it works

Projectile motion separates into independent horizontal (constant velocity) and vertical (constant acceleration −g) components. The simulation integrates with Velocity Verlet and adaptive sub-stepping, and the model is explicit: no air resistance. Without drag, range on flat ground is R = v₀²sin(2θ)/g, maximal at exactly 45°, and complementary angles (30°/60°) give equal range with different heights and flight times. Lowering g (try the Moon preset, g = 1.6 m/s²) stretches both range and flight time by 1/g. The lab closes the predict-verify loop: the closed-form ghost parabola and predicted landing ring are drawn before launch, then the integrated flight lands on them — predicted vs measured range, max height, and flight time in the data panel agree to the integrator's accuracy.

Step-by-step

Set launch speed, angle, height, and gravity (or pick a preset: Optimal 45°, Basketball, Moon Shot, Cannonball).
Every launch first draws the predicted parabola and predicted landing ring, then the flight overlays measured data on the prediction — compare predicted vs measured range, max height, and flight time in the live data panel.
Try complementary angles (30° and 60°) to verify they give the same range, and lower g to see how range scales with 1/g.

Key formulas

$x = v_0\cos\theta \cdot t$ Horizontal position
$y = v_0\sin\theta \cdot t - \frac{1}{2}gt^2$ Vertical position
$R = \frac{v_0^2\sin 2\theta}{g}$ Range (flat ground, no air resistance)

Frequently asked questions

At what angle is range maximum (flat ground)?: R = v₀² sin(2θ)/g is maximum when sin(2θ)=1 → θ=45°.
From x-t data, determine the launch speed if angle=30°.: V_x = v₀cos(30°); slope of x vs t gives v_x; v₀ = v_x/cos(30°).
On the Moon (g = 1.6 m/s²), how does the range of the same launch compare with Earth's?: R = v₀²sin(2θ)/g scales as 1/g: 9.81/1.6 ≈ 6.1× longer. Verify with the Moon Shot preset.