Theory & Mathematics
Understanding the physics behind projectile motion
Model Assumptions
The projectile moves through an ideal vacuum with no drag forces acting upon it.
Gravitational acceleration remains constant throughout the flight, regardless of altitude.
The projectile is treated as a point particle with no rotation or dimensions.
The curvature of the Earth is negligible for the distances considered.
Core Equations
Initial Velocity Components
Where v is the initial speed and θ is the launch angle from horizontal.
Position as a Function of Time
The horizontal position increases linearly with time, while the vertical position follows a parabolic path due to gravity.
Time of Flight
The total time the projectile spends in the air before returning to ground level.
Range (Horizontal Distance)
The horizontal distance traveled when the projectile returns to its initial height. Maximum range occurs at 45°.
Maximum Height
The maximum vertical displacement reached at the apex of the trajectory, where vertical velocity equals zero.
Key Insights
Independence of Motion: The horizontal and vertical motions are independent. The horizontal motion has constant velocity, while the vertical motion is uniformly accelerated.
Symmetry: The trajectory is symmetric about the vertical line through the maximum height. The angle of impact equals the angle of launch.
Parabolic Path: The trajectory follows a parabolic curve described by y = x · tan(θ) − (g · x²) / (2v² · cos²(θ))
Energy Conservation: In the absence of air resistance, mechanical energy is conserved throughout the flight.