Projectile Motion Calculator

Projectile motion describes the curved path of any object launched into the air under gravity alone (no air resistance in the basic model). The motion separates into independent horizontal and vertical components.

Core equations: Horizontal range: R = (v₀² × sin(2θ)) ÷ g

Maximum height: H = (v₀ × sin θ)² ÷ (2g)

Time of flight: T = (2 × v₀ × sin θ) ÷ g

Position at any time t:

• x(t) = v₀ × cos(θ) × t (horizontal, constant velocity)
• y(t) = v₀ × sin(θ) × t − ½ × g × t² (vertical, decelerating under gravity)

Where:

• v₀ = initial launch speed (m/s or ft/s)
• θ = launch angle above horizontal (degrees)
• g = gravitational acceleration = 9.81 m/s² (≈ 32.2 ft/s²)
• sin(2θ) = 2 × sin(θ) × cos(θ)

What each variable means:

• Horizontal component (v₀ × cos θ) — constant; gravity has no horizontal component in the idealized model
• Vertical component (v₀ × sin θ) — decelerates under gravity; reaches zero at maximum height; then accelerates downward
• Optimal angle for maximum range = 45° (when sin(2θ) = sin(90°) = 1, its maximum value)
• Complementary angles — two different angles produce the same range; 30° and 60° both give the same horizontal distance but different heights and times of flight

Reference: trajectory outcomes at different angles (same v₀ = 20 m/s):

• 15°: Range 20.4m, Height 1.3m, Time 0.53s
• 30°: Range 35.3m, Height 5.1m, Time 2.04s
• 45°: Range 40.8m, Height 10.2m, Time 2.89s (maximum range)
• 60°: Range 35.3m, Height 15.3m, Time 3.53s
• 75°: Range 20.4m, Height 19.1m, Time 3.95s

Worked example: Football kicked at v₀ = 25 m/s, θ = 40°.

• sin(40°) = 0.6428, cos(40°) = 0.7660, sin(80°) = 0.9848
• Range = (25² × 0.9848) ÷ 9.81 = (625 × 0.9848) ÷ 9.81 = 615.5 ÷ 9.81 = 62.7 meters
• Height = (25 × 0.6428)² ÷ (2 × 9.81) = (16.07)² ÷ 19.62 = 258.2 ÷ 19.62 = 13.2 meters
• Time = (2 × 25 × 0.6428) ÷ 9.81 = 32.14 ÷ 9.81 = 3.28 seconds