Projectile Time of Flight Calculator

Projectile Time of Flight

The time of flight is how long a projectile remains in the air before landing. This calculator uses the kinematic equation for vertical motion with gravity, including launch from a height above ground level.

Formula

$$t = \frac{v\sin\theta + \sqrt{(v\sin\theta)^2 + 2gh}}{g}$$

Initial velocity $v$ is in m/s, launch angle $\theta$ is measured from horizontal, initial height $h$ is in meters, and $g \approx 9.80665$ m/s². The positive root of the quadratic gives the landing time.

Assumptions

Air resistance is neglected. Launch and landing heights use the same reference level. For level ground ($h = 0$), the formula simplifies to $t = 2v\sin\theta / g$ when the projectile returns to the same height.

Frequently Asked Questions

What launch angle gives maximum range on level ground?

Without air resistance and with launch and landing at the same height, 45° maximizes horizontal range. With a raised launch platform, the optimal angle is slightly lower than 45°.

How does initial height affect flight time?

A positive initial height increases flight time because the projectile has farther to fall. The term $2gh$ under the square root accounts for this extra vertical distance.

Why use only the positive square root?

The negative root would correspond to an unphysical earlier time when the projectile would need to start below ground. The positive root gives the actual landing time.

Does horizontal velocity affect time of flight?

No, in the absence of air resistance. Vertical and horizontal motions are independent, so only the vertical component $v\sin\theta$ and height matter for flight time.

What happens at 0° launch angle?

At 0°, $v\sin\theta = 0$ and the projectile is launched horizontally. Flight time depends only on height: $t = \sqrt{2h/g}$ when launched horizontally from height $h$.

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Time of Flight Projectile Motion