How this instrument works
Time of flight measures how long a projectile stays above the ground between launch and landing, and the formula t = 2v₀sinθ ⁄ g falls straight out of the vertical half of the motion. Gravity acts only downward, so only the vertical component of the launch speed, v₀sinθ, fights against it. That component shrinks to zero at the apex after v₀sinθ ⁄ g seconds, and because gravity is constant and there is no air resistance in this model, the fall back down mirrors the climb exactly — same speed profile, same duration, reversed. Doubling the time to the apex gives the full time aloft.
A mortar crew computing a fuze-time setting, a javelin coach checking whether a throw's angle is wasting hang time instead of distance, a fountain designer timing a water arc to a beat, a golfer's launch monitor reporting hang time on a drive — all of them are reading this same two-term formula, just with the labels swapped. The angle that gives the longest hang time is not the angle that gives the longest range: range peaks at 45°, but time of flight keeps climbing all the way to a dead-vertical 90° shot, since sinθ itself is largest there, even though the projectile then travels no horizontal distance at all.
The formula quietly assumes launch and landing happen at the same height — level ground, a flat fairway, a shooting range. Fire from a cliff edge, a rooftop, or into a valley and the climb and fall are no longer mirror images, so this simple doubling breaks down; that situation needs the fuller kinematic equation, y = v₀sinθ·t − ½gt², solved for the time the height returns to whatever level the target actually sits at. The model also ignores air resistance entirely, which is a fair approximation for a shot put or an artillery shell over a short arc, but increasingly wrong for anything light and fast, like a badminton shuttlecock, where drag cuts the real hang time well below what this vacuum formula predicts.
- Enter the projectile's speed at launch in the Launch speed field, in metres per second.
- Set the Launch angle in degrees, measured up from horizontal; 45° gives the longest range, 90° gives the longest time aloft.
- Read Time of flight in seconds — how long the projectile stays airborne before returning to the launch height.
- Change either field to compare launch conditions; doubling the launch speed alone doubles the time of flight.
Worked example — 30 m/s at a 45° launch angle
Launch speed 30 m/s, launch angle 45°, the classic maximum-range setting on level ground: t = 2 × 30 × sin 45° ⁄ 9.80665 = 4.326289 s. That is roughly four and a third seconds of hang time — plenty long enough for a fielder to judge a fly ball's landing spot, or for a mortar crew's fuze-time table to schedule a burst on arrival.
The same launch speed shows the shape of the formula at its extremes: aimed straight up at 90°, sinθ reaches its maximum of 1 and the flight stretches to 6.118297 s, the longest possible hang time for 30 m/s — but with zero horizontal range, since none of the speed goes sideways. Fired dead level at 0°, sinθ is 0 and the formula returns exactly 0 s, because there is no upward component to carry the projectile before gravity pulls it back to the ground it left.
Questions
Does a bigger launch angle always mean more time in the air?
Up to 90°, yes — time of flight scales with sinθ, which climbs from 0 at a dead-level shot to its maximum of 1 at straight-up. A 30 m/s launch spends 4.326289 s aloft at 45°, but 6.118297 s aloft fired vertically, even though the vertical shot travels no horizontal distance at all.
Isn't 45° also the angle with the longest time of flight?
No — 45° is the angle of maximum range, not maximum time. Time of flight depends only on sinθ, which keeps growing past 45° all the way to 90°, so a near-vertical launch stays up longer than a 45° one even though it lands close to where it started. Range and hang time peak at different angles because range also depends on the horizontal component, cosθ, which shrinks as θ rises.
What if the projectile lands somewhere lower or higher than where it launched?
Then this formula does not apply directly — it assumes the same launch and landing height. A shot fired from a cliff edge, into a valley, or off a rooftop needs the fuller kinematic equation, y = v₀sinθ·t − ½gt², solved for t at whatever height the landing point actually sits, which generally requires the quadratic formula rather than a simple doubling.
Does the calculator account for air resistance?
No, it models ideal projectile motion in a vacuum, with gravity as the only force. That is a fair approximation for dense, compact objects over short arcs — a shot put, an artillery shell — but drag noticeably shortens the real flight time of light or fast objects, like a badminton shuttlecock or a well-hit golf ball at speed.
Where does the value 9.80665 for g come from?
It is standard gravity, the internationally agreed value for Earth's gravitational acceleration, fixed in 1901 and still the reference figure used in engineering and physics calculations worldwide. Local gravity actually varies slightly with latitude and altitude — a touch stronger at the poles, weaker at altitude — but the standard value is accurate enough for any everyday projectile problem.