Trajectory Calculator

A trajectory is the path that a projectile follows through space after being launched. In physics, projectile motion combines both horizontal and vertical motion, resulting in a parabolic path when air resistance is neglected.

Key components of projectile motion include:

• Horizontal motion at constant velocity
• Vertical motion affected by gravity
• Independence of horizontal and vertical components
• Parabolic path shape

The trajectory calculation involves several steps:

1. Decompose initial velocity into components:
   • v₀ₓ = v₀ × cos(θ)
   • v₀ᵧ = v₀ × sin(θ)
2. Calculate time of flight:
   • t = (v₀ᵧ + √(v₀ᵧ² + 2gh₀)) / g
3. Determine maximum height and range:
   • h_max = h₀ + v₀ᵧ²/(2g)
   • Range = v₀ₓ × t
4. Plot trajectory points using parametric equations:
   • x(t) = v₀ₓt
   • y(t) = h₀ + v₀ᵧt - (gt²)/2

Trajectory calculations are essential in many fields:

• Sports: Optimizing ball trajectories in games like basketball, golf, and baseball
• Military: Artillery targeting and missile guidance systems
• Space Exploration: Planning spacecraft trajectories and orbital maneuvers
• Engineering: Designing launch systems and projectile-based equipment
• Education: Teaching physics concepts and demonstrating projectile motion

• Initial Velocity: Higher velocities result in greater ranges and heights
• Launch Angle: Different angles optimize for either maximum height or range
• Initial Height: Starting height affects total time of flight and range
• Gravity: Determines the vertical acceleration of the projectile
• Air Resistance: Real-world factor that reduces range and height (not included in this ideal model)