Projectile Motion Calculator
g = 9.81 m/s²
Try an example
Range
40.77 m
Max Height
10.19 m
Flight Time
2.88 s
Trajectory
Formulas
R = v₀² sin(2θ) / g = 40.77 m
h = v₀² sin²(θ) / (2g) = 10.19 m
T = 2 v₀ sin(θ) / g = 2.88 s
Calculate the range, maximum height and flight time of a projectile launched from ground level at a given speed and angle, ignoring air resistance, and see its trajectory.
Formula
Worked Example
At 45°, sin(2θ) = 1, so the range is 20² ÷ 9.81 = 40.77 m, the longest possible at this speed on level ground. The vertical speed of 20 × sin 45° = 14.14 m/s takes 1.44 s to fall to zero, so the flight lasts twice that.
Frequently Asked Questions
Why does 45° give the maximum range?
The range depends on sin(2θ), which peaks at 1 when 2θ = 90°. Without air resistance, angles that add up to 90°, such as 30° and 60°, land at the same distance, the steeper one after a higher, longer flight.
How does air resistance change the result?
Drag shortens the range and makes the path asymmetric, falling more steeply than it rose, and the best angle drops below 45°: often 35–42° for balls in sports. These formulas are accurate for slow, dense objects over short distances.
What if the projectile is launched from a height?
Then it stays in the air longer and goes farther, and the optimal angle is below 45°. Solve y = h + v₀ sin(θ) t − ½gt² = 0 for the flight time, then multiply by v₀ cos(θ) for the range.
Common Mistakes
- !Using degrees instead of radians — trigonometric functions require radians in most contexts; this calculator converts automatically.
- !Ignoring air resistance — real projectiles experience drag, so actual range is shorter than the ideal calculation.
- !Assuming a 45° launch angle always gives maximum range — this is true in a vacuum, but real-world conditions (air resistance, launch height) change the optimal angle.