Trajectory Calculator

Calculate projectile motion trajectories with detailed physics analysis. Perfect for students, engineers, and game developers.

Projectile Motion Equations:

Horizontal position: x(t) = v₀·cos(θ)·t

Vertical position: y(t) = v₀·sin(θ)·t - ½·g·t²

Range: R = v₀²·sin(2θ)/g | Max height: H = (v₀·sin(θ))²/(2g) | Flight time: T = 2·v₀·sin(θ)/g

Units
m/s
Speed at launch
degrees
Angle above horizontal (0-90°)
m
Height above ground at launch
m/s²
Acceleration due to gravity

0 = no air resistance (simplified model)
m
Height of landing surface (for impact calculation)
Cannon Shot (45°, 50 m/s)
Basketball (50°, 10 m/s)
Golf Drive (12°, 70 m/s)
Arrow (30°, 60 m/s)
Baseball (35°, 40 m/s)
On Moon (45°, 20 m/s)
Calculating trajectory...

Projectile Motion Physics

Projectile motion refers to the motion of an object projected into the air at an angle, subject only to the acceleration of gravity. The path followed by a projectile is called its trajectory, which is always parabolic (in the absence of air resistance).

Key Assumptions in Projectile Motion:

  • Acceleration due to gravity is constant (9.81 m/s² downward)
  • Air resistance is negligible (unless specified)
  • The Earth's curvature is negligible over short ranges
  • The projectile's mass does not affect its trajectory

Projectile Motion Equations

Parameter Equation Explanation
Horizontal Position x(t) = v₀·cos(θ)·t Constant horizontal velocity (no acceleration in x-direction)
Vertical Position y(t) = v₀·sin(θ)·t - ½·g·t² Vertical motion with constant downward acceleration
Time of Flight T = (v₀·sin(θ) + √((v₀·sin(θ))² + 2·g·h₀)) / g Time from launch to landing (for h₀ > 0)
Range R = v₀·cos(θ)·T Horizontal distance traveled
Maximum Height H = h₀ + (v₀·sin(θ))² / (2·g) Highest point in the trajectory (when vᵧ = 0)
Trajectory Equation y = x·tan(θ) - (g·x²) / (2·v₀²·cos²(θ)) Parabolic path (eliminating time parameter)

Optimal Launch Angles

1

Maximum Range: For a given initial velocity and no air resistance, the maximum range is achieved at a launch angle of 45° (when initial height = 0).

2

Complementary Angles: Launch angles that add up to 90° (e.g., 30° and 60°) produce the same range (when initial height = 0).

3

Height Considerations: When launching from a height above the landing point, the optimal angle for maximum range is less than 45°.

Applications of Trajectory Calculations

  • Sports: Analyzing basketball shots, golf drives, baseball hits
  • Engineering: Designing water fountains, fireworks displays
  • Military: Calculating artillery shell trajectories
  • Game Development: Creating realistic projectile physics in video games
  • Safety: Determining safe zones for fireworks or construction sites

Calculator Features:

  • Calculates all key trajectory parameters: range, max height, flight time
  • Supports SI and Imperial units with automatic conversion
  • Includes option for initial height above ground
  • Generates detailed trajectory graph with key points marked
  • Provides step-by-step calculations for educational purposes

Frequently Asked Questions

The range equation R = (v₀²·sin(2θ))/g shows that range is maximized when sin(2θ) is maximized. Since sin(90°) = 1 is the maximum value of the sine function, 2θ = 90°, thus θ = 45°. This assumes no air resistance and launch from ground level.

Air resistance reduces both the range and maximum height of a projectile. It also causes the trajectory to become asymmetric - the descending part of the path is steeper than the ascending part. For objects with high speed or large surface area, air resistance can significantly alter the trajectory from the ideal parabolic path.

When launching from above ground level, the projectile has more time to travel horizontally before hitting the ground, increasing the range. The optimal angle for maximum range becomes slightly less than 45°, depending on the initial height and velocity.

Yes, you can change the gravity value to simulate conditions on different celestial bodies. For example, use 1.62 m/s² for the Moon, 3.71 m/s² for Mars, or 24.79 m/s² for Jupiter. The calculator uses your specified gravity value in all calculations.

For many applications (sports, basic engineering, education), these calculations provide excellent approximations. However, for precision applications (artillery, space missions), additional factors must be considered: air resistance (which varies with velocity), wind, Earth's rotation (Coriolis effect), and variations in gravitational acceleration with altitude.