Projectile Motion Calculator
Enter the launch speed, the angle above the horizontal and the height of the launch point (0 for ground level). Air resistance is ignored and gravity is 9.80665 m/s².
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Formula
With launch speed v, angle θ above horizontal, launch height h₀ and g = 9.80665 m/s² (1 ft = 0.3048 m for imperial):
vx = v·cos θ,vy = v·sin θ- Flight time:
T = (vy + √(vy² + 2·g·h₀)) ÷ g - Range:
R = vx · T - Maximum height:
H = h₀ + vy² ÷ (2·g) - Time to the highest point:
vy ÷ g - Impact speed:
√(v² + 2·g·h₀)
Air resistance, wind and the curvature of the Earth are ignored. The angle runs from 0° (horizontal) to 90° (straight up).
Worked example
Launch at 20 m/s and 45° from the ground. vy = 14.142 m/s, so T = 2 × 14.142 ÷ 9.80665 = 2.884 s, range = 14.142 × 2.884 = 40.79 m, and H = 14.142² ÷ (2 × 9.80665) = 10.20 m.
From a 10 m high launch at 30°: T = 2.774 s, range 48.06 m, maximum height 15.10 m, impact speed 24.42 m/s.
Frequently asked questions
What is the range of a projectile?
From ground level, range = v² sin(2θ) / g. At 20 m/s and 45° that is 400 × 1 ÷ 9.80665 = 40.79 m. With a raised launch the calculator uses the full flight time instead.
Which angle gives the longest range?
45° from level ground, because sin(2θ) is largest there. From a height the best angle is below 45°: launching at 20 m/s from 10 m gives 49.10 m at 45° but 49.78 m at 40°.
How do I find the maximum height?
H = h₀ + (v sin θ)² / 2g. At 20 m/s and 45° from the ground, (14.142)² ÷ (2 × 9.80665) = 10.20 m.
Why is the real distance shorter?
Air drag slows real projectiles, and spin and wind change the path. This calculator models the ideal case with only gravity, which is accurate for dense, slow objects over short distances.