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Projectile Motion

Beginner

Explore the parabolic trajectories of objects launched at various angles and velocities. Understand how gravity shapes the path of projectiles and learn to predict range, maximum height, and flight time.

Key Formulas

Range
R=v02sin⁡(2θ)gR = \frac{v_0^2 \sin(2\theta)}{g}

Horizontal distance traveled by a projectile launched from ground level.

Maximum Height
H=v02sin⁡2(θ)2gH = \frac{v_0^2 \sin^2(\theta)}{2g}

The peak altitude reached during projectile flight.

Time of Flight
T=2v0sin⁡(θ)gT = \frac{2v_0 \sin(\theta)}{g}

Total time the projectile remains airborne before returning to launch height.

Horizontal Position
x(t)=v0cos⁡(θ)⋅tx(t) = v_0 \cos(\theta) \cdot t

Horizontal displacement as a function of time.

Vertical Position
y(t)=v0sin⁡(θ)⋅t−12gt2y(t) = v_0 \sin(\theta) \cdot t - \frac{1}{2}g t^2

Vertical displacement as a function of time.

Key Concepts

  • Independence of horizontal and vertical motion
  • Constant horizontal velocity (no air resistance)
  • Constant gravitational acceleration
  • Symmetric parabolic trajectory
  • Optimal launch angle of 45 degrees for maximum range
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