🎯 Projectile Motion Calculator

Simulate projectile trajectories with real-time animation

Trajectory Simulation Range & Height Animated
🎯 Projectile Motion Simulator

Set the initial velocity and launch angle, then click Launch! Watch the projectile follow its parabolic path. 45° gives maximum range on flat ground.

Launch Settings

Configure your projectile

🚀 Initial Velocity (v₀)
📐 Launch Angle (θ)
45°
🌍 Gravity (g)
📏
Range
40.8 m
⬆️
Max Height
10.2 m
⏱️
Flight Time
2.88 s
📐 Display Units

📚 Try These

⚽
Soccer Kick
25 m/s @ 35°
🏀
Basketball
8 m/s @ 55°
🎾
Tennis Serve
50 m/s @ 10°
💣
Cannonball
80 m/s @ 45°

🎬 Trajectory Simulation

🎯
0 m 40 m
Position
x: 0 m, y: 0 m
Velocity
vx: 0, vy: 0 m/s
Time
t = 0.00 s
Max: 10.2 m

📊 Motion Graphs

📈 x vs t
📈 y vs t
📈 y vs x (Trajectory)

📝 Projectile Motion Formulas

R = v₀²sin(2θ) / g Range (horizontal distance)
H = v₀²sin²(θ) / 2g Maximum height
T = 2v₀sin(θ) / g Total flight time
🧮 Step-by-Step Solution ▼ Show

Understanding Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject only to gravity. The path of a projectile is called its trajectory, and it always forms a parabola.

Key Concepts

  • Independence of motion: Horizontal and vertical motions are independent of each other
  • Horizontal motion: Constant velocity (no acceleration) - x = v₀cos(θ) × t
  • Vertical motion: Accelerated by gravity - y = v₀sin(θ) × t - ½gt²
  • Optimal angle: 45° gives maximum range on flat ground
  • Complementary angles: Angles like 30° and 60° give the same range

The Physics Behind It

When you launch a projectile at angle θ with initial velocity v₀, the velocity has two components:

  • Horizontal component: v₀ₓ = v₀ × cos(θ) - stays constant throughout flight
  • Vertical component: v₀ᵧ = v₀ × sin(θ) - decreases going up, increases coming down

Why 45° is Optimal

The range formula R = v₀²sin(2θ)/g is maximized when sin(2θ) = 1, which occurs at 2θ = 90°, or θ = 45°. At this angle, you get the perfect balance between horizontal distance and time in the air.

Real-World Applications

  • Sports: Soccer kicks, basketball shots, golf drives, javelin throws
  • Military: Artillery trajectories, missile paths
  • Engineering: Water fountains, sprinkler systems
  • Space: Rocket launches, satellite deployments