Projectile Range
Cannon with adjustable angle and speed launches a projectile — live readouts of range, max height, and flight time.
Objective
Verify that horizontal range follows R = v₀²·sin(2θ)/g, confirming that 45° maximizes range for any fixed speed and that complementary angles (e.g. 30° and 60°) produce equal ranges. The model treats the projectile as a point mass launched from ground level with no air resistance.
Setup
- Set the launch angle slider to 45° and the initial speed slider to 25 m/s — these are the defaults. Observe the dashed ghost arc showing the predicted analytical trajectory before launch.
- Press Start and watch the projectile follow the parabolic path. When it lands, record the Range readout (≈ 63.8 m), the Height readout (≈ 15.9 m), and the Time readout (≈ 3.60 s).
- Press Reset. Change the angle to 30° (keep speed at 25 m/s). Press Start and record the range. Then Reset, set angle to 60°, and press Start — compare the two range readouts to confirm they match.
- Press Reset. Set angle back to 45° and increase speed to 50 m/s. Press Start and record the new range — it should be approximately four times the range at 25 m/s, confirming the v₀² dependence.
Analytical Prediction
The range formula gives R = v₀² · sin(2θ) / g. With v₀ = 25 m/s and θ = 45°:
Flight time T = 2 · v₀ · sin(θ) / g:
For complementary angles at 30° and 60°: sin(2 × 30°) = sin 60° ≈ 0.866 and sin(2 × 60°) = sin 120° ≈ 0.866 — identical, so both give R ≈ 55.2 m. At v₀ = 50 m/s and θ = 45°: R ≈ 254.8 m — exactly four times the 25 m/s result.
Results Analysis
After each run, compare the Range readout (#xOut) to the analytical prediction. At v₀ = 25 m/s and θ = 45°, the readout should show approximately 63.7–63.8 m and the Time readout (#tOut) approximately 3.59–3.60 s. The Height readout (#hOut) tracks peak height — check it against H = v₀²·sin²(θ)/(2g) ≈ 15.9 m. At 30° and 60°, confirm both Range readouts agree to within 0.2 m. At v₀ = 50 m/s, the Range readout should be approximately 254–255 m, consistent with the four-fold scaling from the v₀² dependence.
Source of Error
This model omits air resistance (drag), the finite size of the projectile (point-mass idealization), Earth's curvature, and any spin or Magnus effect. Both the analytical prediction and the simulation assume identical idealizations — a point mass, uniform gravity g = 9.81 m/s², flat horizontal ground, and launch from ground level with zero initial height. These idealizations cancel when comparing prediction to readout rather than contributing to the residual. Any small discrepancy between the predicted value and the displayed readout is therefore purely numerical, not physical.
Further Exploration
- What angle produces the same range as 30°? Sweep the angle slider while watching the Range readout — can you identify the second angle without calculating it first? What is the mathematical relationship between the two angles?
- Double the speed from 25 m/s to 50 m/s while holding the angle at 45°. By what factor does the range change? Does the ratio match the v₀² prediction? Repeat the experiment at θ = 30° to confirm the scaling holds at other angles.
- Sweep the angle slowly from 5° to 85° at a fixed speed of 25 m/s. At what angle does the Height readout (#hOut) reach its maximum? Is that the same angle that maximizes the Range readout (#xOut)?
- Set speed to 10 m/s (the minimum) and angle to 45°. Compare the ghost arc to the live trail — do they match closely? Now increase speed to 50 m/s. Does the agreement between the arc and trail change?