Angled Projectile Motion
Overview
When launched at angle θ above horizontal with speed v₀, decompose: Horizontal: v_x = v₀cosθ (constant throughout). Vertical: v_y = v₀sinθ − gt (changes). Time to max height: t_top = v₀sinθ/g (set v_y = 0). Maximum height: H = (v₀sinθ)²/(2g). Horizontal range (flat ground): R = v₀²sin(2θ)/g. Range is maximum at θ = 45°. Complementary angles give equal ranges (e.g. 30° = 60°). At any point: total speed v = √(v_x² + v_y²). At the top: v_y = 0, so speed = v_x = v₀cosθ.
Decomposing the initial velocity
When a projectile is launched at angle θ above the horizontal with initial speed v₀, split it into components: Horizontal: v_x = v₀ cos θ (constant throughout). Vertical: v_y = v₀ sin θ − gt (decreases going up, becomes negative on the way down). At any time t: position x = v₀ cos θ · t and y = v₀ sin θ · t − ½gt².
Maximum height and range
At maximum height, v_y = 0. Time to reach top: t_top = v₀ sin θ / g. Maximum height: H = (v₀ sin θ)² / (2g). For a flat landing at the same height, time of flight = 2t_top. Horizontal range: R = v₀ cos θ × 2t_top = v₀² sin(2θ) / g. The range is maximised at θ = 45°. Complementary angles (e.g. 30° and 60°) give equal ranges.
- ⚠Using the range formula R = v₀² sin(2θ)/g when the landing height ≠ launch height — it only works on flat ground
- ⚠Forgetting that at the top, the horizontal velocity component is still v₀ cos θ (not zero)
- ⚠Not doubling t_top to get total flight time