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Projectile Motion

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🎯 About Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The horizontal and vertical motions are independent of each other — a revolutionary insight from Galileo. The resulting path is a parabola (without air resistance).

💡 Key Physics Concepts

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    Independence of Motions: Horizontal velocity is constant; vertical motion is free fall
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    Optimal Angle: 45° gives maximum range (no air resistance)
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    Complementary Angles: Angles θ and (90°-θ) give the same range
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    Air Resistance: Drag force F = -½ρCdA|v|v opposes motion, making trajectory asymmetric
  • •
    Maximum Height: Occurs when vertical velocity = 0

📐 Formulas

Horizontal position:x(t) = v₀·cos(θ)·t
Vertical position:y(t) = v₀·sin(θ)·t - ½gt²
Horizontal velocity:vₓ = v₀·cos(θ)
Vertical velocity:vᵧ = v₀·sin(θ) - gt
Range:R = v₀²·sin(2θ) / g
Max Height:H = v₀²·sin²(θ) / (2g)
Time of Flight:T = 2v₀·sin(θ) / g
With air resistance (drag):F_drag = -C_d · |v| · v̂

💨 Air Resistance

In reality, projectiles experience air resistance proportional to the square of their speed (quadratic drag). This simulation uses RK4 numerical integration to solve the equations of motion with drag. You'll notice: shorter range, asymmetric trajectory (steeper descent), and reduced time of flight compared to the ideal case.

🔬 Real-World Applications

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    Sports: Basketball, football, golf, javelin — all follow projectile physics
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    Military: Artillery trajectory calculation and ballistic missile design
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    Space: Rocket launch trajectories and orbital insertion
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    Engineering: Water fountains, fireworks, sprinkler systems
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    Entertainment: Angry Birds physics engine!

🎮 How to Use the Simulation

  • 1.
    Set velocity and angle, then hit 🚀 Launch
  • 2.
    Watch the ghost markers appear every 0.5s to see time spacing
  • 3.
    Enable Air Resistance to see how drag affects the trajectory
  • 4.
    Compare predicted vs actual range after landing
  • 5.
    Try different gravity presets — the Moon gives crazy range!
  • 6.
    Adjust target distance and try to hit the bullseye
  • 7.
    Use Speed slider for slow-motion analysis
🎯Launch the Projectile Experiment