Key Equations (No Air Resistance)
Horizontal motion (constant velocity):
vₓ = v₀ cos(θ) | x = vₓ t
Vertical motion (constant acceleration, g downward):
vᵧ = v₀ sin(θ) - gt | y = h₀ + v₀ sin(θ) t - ½gt²
Derived Quantities
Time of flight: T = (v₀ sin(θ) + √(v₀²sin²θ + 2gh₀)) / g
Max height: H_max = h₀ + v₀²sin²θ / (2g)
Range: R = v₀ cos(θ) × T
Velocity at apex: v_apex = v₀ cos(θ) (only horizontal)
Impact speed: v_impact = √(vₓ² + vᵧ²) where vᵧ = -√(v₀²sin²θ + 2gh₀)
Key Insights
- Horizontal velocity vₓ remains constant throughout (no air resistance)
- At apex, vertical velocity vᵧ = 0, so velocity is purely horizontal
- For h₀ = 0, maximum range occurs at θ = 45° (complementary angles give same range)
- Time of flight depends only on vertical motion