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) stays constant for the whole flight, because gravity has no sideways pull in the idealized model
• Vertical component (v₀ × sin θ) decelerates on the way up, reaches zero at the peak, then accelerates back down
• Optimal angle for maximum range = 45° (when sin(2θ) = sin(90°) = 1, its maximum value)
• Complementary angles: two launch angles give the same range. 30° and 60° land in the same spot, but the 60° shot climbs three times higher and stays in the air 73% longer

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

• 15°: Range 20.4m, Height 1.4m, Time 1.06s
• 30°: Range 35.3m, Height 5.1m, Time 2.04s
• 45°: Range 40.8m, Height 10.2m, Time 2.88s (maximum range)
• 60°: Range 35.3m, Height 15.3m, Time 3.53s
• 75°: Range 20.4m, Height 19.0m, Time 3.94s

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