Projectile Motion
Parabolic trajectory under gravity. Horizontal: constant v; vertical: uniform acceleration g. R = v₀²sin(2θ)/g; 45° gives max range.
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45° launch gives maximum range (ideal) Range proportional to v₀² Time of flight T = 2v₀sinθ/g Trajectory is parabola
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Why: Projectile motion models sports, ballistics, and any object under gravity. 45° gives maximum range (no air resistance).
How: Decompose into vx = v₀cosθ (constant), vy = v₀sinθ − gt. Range from R = v₀²sin(2θ)/g.
Run the calculator when you are ready.
⚙️ Launch Parameters
Projectile Settings
Gravitational Environment
Quick Angle Selection
📊 Projectile Motion Results
📈 Trajectory Analysis Dashboard
🎯 Projectile Trajectory
Height vs horizontal distance with apex and impact points marked
📏 Height vs Time
🚀 Velocity Components vs Time
📐 Range vs Launch Angle
⚡ Energy Distribution
📝 Step-by-Step Solution
Initial Velocity: 30.00 m/s at 45.0°
Initial Height: 0.00 m
Gravitational Acceleration: 9.81 m/s² (Earth)
Horizontal velocity component: vₓ = v₀ × cos(θ)
vₓ = 30.00 × cos(45.0°)
→ vₓ = 21.2132 m/s
Vertical velocity component: vᵧ = v₀ × sin(θ)
vᵧ = 30.00 × sin(45.0°)
→ vᵧ = 21.2132 m/s
Time to reach maximum height: t_max = vᵧ₀ / g
t_max = 21.21 / 9.81
→ t_max = 2.1624 seconds
Maximum height: h_max = h₀ + vᵧ₀² / (2g)
h_max = 0.00 + 21.21² / (2 × 9.81)
→ h_max = 22.9358 meters
Time of flight: solving h₀ + vᵧ₀t - ½gt² = 0
t = (vᵧ₀ + √(vᵧ₀² + 2gh₀)) / g
→ t = 4.3248 seconds
Horizontal range: R = vₓ × t
R = 21.21 × 4.32
→ R = 91.7431 meters
Impact velocity magnitude:
v_impact = √(vₓ² + vᵧ²)
→ v_impact = 30.0000 m/s
Impact angle (below horizontal):
→ θ_impact = 45.00°
Initial Kinetic Energy: KE₀ = ½mv₀²
KE₀ = 0.5 × 1.00 × 30.00²
→ KE₀ = 450.0000 J
Maximum Potential Energy (at apex):
→ PE_max = 225.0000 J
Impact Kinetic Energy:
→ KE_impact = 450.0000 J
📖 What is Projectile Motion?
Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The key insight is that horizontal and vertical components of motion are independent of each other.
🎯 Key Principles
- • Horizontal velocity (vₓ) remains constant
- • Vertical velocity (vᵧ) changes due to gravity
- • Path forms a parabola (without air resistance)
- • 45° gives maximum range (from ground level)
🏀 Real Applications
- • Sports: basketball, golf, football
- • Military: artillery, missiles
- • Entertainment: fireworks, fountains
- • Engineering: irrigation, robotics
🧮 Projectile Motion Formulas
Velocity Components
Position Equations
Key Results (from ground level, h₀ = 0)
Maximum Height:
h_max = (v₀ sin θ)² / (2g)
Range:
R = v₀² sin(2θ) / g
Time of Flight:
T = 2 × v₀ sin(θ) / g
🎯 Launch Angle Optimization
Maximum Range: 45°
For ground-to-ground projectiles, 45° maximizes range because sin(2×45°) = sin(90°) = 1.
- • Angles 30° and 60° give equal range
- • Lower angles = flatter trajectory, faster
- • Higher angles = higher arc, more hangtime
Special Cases
- • Elevated launch: Optimal angle < 45°
- • Uphill target: Optimal angle > 45°
- • With air resistance: Optimal ≈ 35-40°
- • Sports specific: Varies by goal (height vs distance)
🏆 Sports Projectile Reference
| Sport | Velocity | Typical Angle | Max Range |
|---|---|---|---|
| 🏀Basketball Shot | 7.5 m/s | 52° | 5.6 m |
| ⚾Baseball Pitch | 40 m/s | 0° | N/A m |
| ⛳Golf Ball | 70 m/s | 12° | 203.2 m |
| 🏈Football Pass | 25 m/s | 40° | 62.7 m |
| 🏹Arrow | 80 m/s | 5° | 113.3 m |
| ⚽Soccer Kick | 30 m/s | 45° | 91.7 m |
| 🎾Tennis Serve | 60 m/s | -5° | N/A m |
| 🥇Shot Put | 14 m/s | 42° | 19.9 m |
🔗 Related Calculators
For educational and informational purposes only. Verify with a qualified professional.
🔬 Physics Facts
R = v₀²sin(2θ)/g; max at 45°
— NIST
H_max = v₀²sin²θ/(2g)
— MIT
Time of flight T = 2v₀sinθ/g
— NASA
g varies: Earth 9.81, Moon 1.62 m/s²
— Physics
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