Time of Flight (Projectile Motion)

Because vertical and horizontal motion are completely independent. Gravity acts only in the vertical direction, so it can only change the vertical component of velocity. The horizontal component is unaffected by gravity and continues at a constant v·cosθ throughout the flight. The projectile stays aloft until gravity has cancelled and then reversed the vertical component of the launch velocity, and how long that takes depends only on v·sinθ and g. The horizontal distance covered during that time depends on v·cosθ, but the time itself does not. This independence is the foundation of projectile theory and one of Galileo's most important insights.

Because range and flight time depend on different combinations of the angle. Range depends on sin(2θ), which peaks at 2θ = 90°, that is, at θ = 45°. Flight time depends on sinθ, which peaks at θ = 90° — straight up. So a 45° launch is the optimal compromise between getting enough height to stay in the air and enough horizontal velocity to cover ground, while a 90° launch stays up longest but comes straight back down with zero range. A 30° and a 60° launch have exactly the same range (since sin 60° = sin 120°), but the 60° shot stays in the air longer — about 73 % longer — because it climbs higher before falling back.

Flight time is inversely proportional to gravity: halve g and the projectile stays in the air twice as long. This is why a golf ball hit on the Moon travels six times further and hangs for six times longer than the same shot on Earth — the Moon's gravity is only 1.62 m/s² compared with Earth's 9.807 m/s². The Apollo 14 astronauts demonstrated this with a golf club and a spare club head: the ball was reported to travel "miles and miles" (in reality about 200 m, but that's still over ten times a normal drive). On the Sun, where g = 274 m/s², the same shot would be airborne for a fraction of a second and travel just a few metres.

The formula t = 2v·sinθ/g assumes four things. First, that air resistance is negligible, which is true only for dense, fast, streamlined projectiles over short distances. Second, that the gravitational acceleration g is constant throughout the flight, which is accurate for heights much smaller than the Earth's radius. Third, that the Earth is flat over the range of the projectile, which fails beyond a few tens of kilometres. Fourth, that the launch and landing heights are the same — if the projectile lands at a different height, the formula becomes a quadratic equation and the flight time changes. For a ball thrown off a cliff, the descent takes longer; for a projectile fired upward from a platform, both the ascent and descent times differ from the symmetric case.

Substantially, for anything light, blunt or slow. A soccer ball kicked at 25 m/s and 45° has a vacuum flight time of 3.605 s and range of 63.7 m, but in reality it stays up for about 3.0 s and travels around 45 m — a 17 % reduction in time and 30 % in range. A table-tennis ball, by contrast, is so light that the same vacuum calculation would be off by an order of magnitude. Air resistance is roughly proportional to v² for a bluff body, so it matters most at high speeds and for objects with a large frontal area relative to their mass. The only way to get accurate real-world predictions is to solve the equations of motion numerically with a drag model appropriate to the projectile.

Not directly. The formula assumes the projectile returns to the same height it was launched from. If you launch from a height h above the landing point, the flight time is given by the solution of the quadratic equation h = −v·sinθ·t + ½g·t², which gives t = (v sin θ + √(v² sin²θ + 2gh)) / g. When h = 0 this reduces to the symmetric result t = 2v·sinθ/g. If you launch upward from a cliff of height h, the extra term √(v²sin²θ + 2gh) makes the flight time longer; if you launch horizontally (θ = 0) from the same cliff, the flight time is simply √(2h/g), the free-fall time. For those cases you need the more general form, not the symmetric formula used here.

In casual use, yes — "hang time" and "time of flight" both refer to how long a projectile stays in the air. In sport analytics, hang time is usually the measured time from launch to landing of a punt, kick or jump, and it is essentially the same quantity. In physics, the term "time of flight" is preferred because it is more precise. Note that in American football, the "hang time" of a punt is often reported in the broadcast as a key statistic because it gives the coverage team time to get downfield; punters aim for 4.5–5.5 s, which corresponds to a launch speed of about 28–34 m/s at 55–65°. The formulas here let you work backwards from a measured hang time to the launch speed, which is how punt-tracking systems do it.