Time of Flight Calculator – Projectile Motion
With this time of flight calculator, you can easily calculate the time a projectile remains in the air. Enter the initial velocity, launch angle, and optional initial height to find out exactly how long the object stays airborne.
Time of Flight Equation
When you throw a ball or fire an arrow at an angle, it follows a parabolic path before hitting the ground. The time of flight depends on the initial velocity, the launch angle, and the height from which it is launched.
Case 1 — Launching from the Ground (Initial Height = 0)
Starting with the vertical position equation:
y = V₀ · t · sin(α) − ½ · g · t²
Where:
- V₀ — Initial velocity
- t — Time since the start of flight
- α — Launch angle above horizontal
- g — Gravitational acceleration (9.81 m/s² or 32.174 ft/s²)
The flight ends when y = 0 (the projectile returns to ground level). Setting y = 0 and solving for t gives the total flight time:
t = 2 · V₀ · sin(α) / g
Case 2 — Launching from an Elevated Position (Initial Height > 0)
When the projectile is launched from height h above the landing point, the time of flight is found by solving the quadratic equation:
0 = h + V₀ · sin(α) · t − ½ · g · t²
Taking the positive root:
t = (V₀ · sin(α) + √(V₀² · sin²(α) + 2 · g · h)) / g
Time of Flight — Worked Examples
Example 1 — Ball Thrown at 45° (Metric)
A ball is thrown at 20 m/s at a 45° angle from ground level.
- Vy₀ = 20 · sin(45°) = 14.142 m/s
- t = 2 × 14.142 / 9.81 ≈ 2.884 s
Example 2 — Arrow Shot at 30° (Imperial)
An arrow is launched at 200 ft/s at a 30° angle from ground level.
- Vy₀ = 200 · sin(30°) = 100 ft/s
- t = 2 × 100 / 32.174 ≈ 6.215 s
Example 3 — Projectile from a Cliff (Elevated Launch)
A projectile is fired at 15 m/s at 60° from a cliff 10 m high.
- Vy₀ = 15 · sin(60°) = 12.990 m/s
- t = (12.990 + √(12.990² + 2 × 9.81 × 10)) / 9.81 ≈ 3.383 s
Which Launch Angle Gives the Longest Flight Time?
Looking at the ground-launch formula t = 2 · V₀ · sin(α) / g, the time of flight is maximized when sin(α) is at its maximum — which occurs at α = 90° (a straight vertical throw). The higher the sine value, the longer the object remains in the air.
Note that this differs from the optimal angle for maximum horizontal range, which is 45°. If you want to maximize the distance traveled, use our Projectile Range Calculator.
Free Fall as a Special Case
When the initial velocity is zero and the object is dropped from height h, this reduces to the free fall problem:
t = √(2h / g)
You can explore this further with our Free Fall Calculator.
Units Supported
- Metric (SI): velocity in m/s, height in m, g = 9.81 m/s²
- Imperial (US): velocity in ft/s, height in ft, g = 32.174 ft/s²
FAQs
Does air resistance affect the time of flight?
This calculator assumes no air resistance (vacuum). In real-world conditions, air drag reduces the time of flight slightly, especially for fast or light projectiles.
Can the angle be 0°?
An angle of 0° (horizontal launch) with zero initial height means the object immediately hits the ground, so the calculation requires a positive initial height. If you want to calculate a purely horizontal launch from a height, enter the height and set the angle to 0.
What is the time of flight for a vertical throw?
For a vertical throw (α = 90°), the formula simplifies to t = 2 · V₀ / g. For example, a ball thrown straight up at 10 m/s will be in the air for about 2.04 seconds.
How is time of flight related to maximum height?
The projectile reaches its maximum height at exactly t/2 (half the total flight time, for ground-to-ground launches). The maximum height is h_max = Vy₀² / (2g) above the launch point.