Projectile Motion Simulator

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The path that the object follows is called its trajectory.

Where:

• y is the vertical position (in meters)
• y₀ is the initial height (in meters)
• v₀ is the initial velocity (in m/s)
• g is acceleration due to gravity (-9.81 m/s²)
• t is time (in seconds)

• Horizontal Motion: x = x₀ + v₀ₓt
  • Constant velocity (no acceleration)
  • v₀ₓ = v₀cos(θ)
• Vertical Motion: y = y₀ + v₀ᵧt + ½gt²
  • Accelerated motion due to gravity
  • v₀ᵧ = v₀sin(θ)
• Time of Flight: t = 2v₀sin(θ)/g
  • Time until projectile returns to initial height
  • Only valid when y = y₀
• Maximum Height: h = v₀²sin²(θ)/2g
  • Highest point in the trajectory
  • Occurs when vertical velocity is zero
• Range: R = v₀²sin(2θ)/g
  • Horizontal distance traveled
  • Maximum when θ = 45°

• Independence of Motion:
  • Horizontal and vertical motions are independent
  • Horizontal velocity remains constant
  • Vertical velocity changes due to gravity
• Parabolic Path:
  • Trajectory forms a parabola
  • Symmetrical about the peak
  • Time up equals time down
• Optimal Angle:
  • 45° gives maximum range
  • Complementary angles give same range
  • 90° gives maximum height

• Sports (basketball, football, golf)
• Artillery and ballistics
• Water fountains and sprinklers
• Fireworks displays
• Spacecraft trajectories
• Water slides and amusement park rides