Time of flight (vertical)

Intermediate

Up and down takes twice the rise time.

Key relationT = 2u/g

For a throw up and return to the starting height, total time T = 2u/g. If it lands at a different height solve s = ut − ½gt² as a quadratic, taking the positive root.

Worked example

Thrown at 15 m/s from the ground: T = 2·15/9.8 ≈ 3.06 s.

Common mistake

A quadratic has two roots; one is often negative or before launch.

In more detail

The time of flight for a vertical throw is the time spent in the air from launch until it returns to the same height. For a launch and return to the same level, the time is 2u/g.

How to solve it

For a return to the launch height, the total time of flight is T = 2u/g.
For a point at height h on the way up or down, solve h = ut − ½gt², a quadratic in t.
Take both roots of the quadratic: one on the way up, and one on the way down.
Check each root by substituting it back, and reject any that do not make physical sense.

Worked examples

Example 1: A ball is launched upward at 12 m/s and caught at the same height. How long is it in the air?

T = 2u/g = 2 × 12 / 9.8 = 2.45 s.

Example 2: The same ball passes a point 5 m above the launch point. At what two times does this happen?

5 = 12t − 4.9t², so 4.9t² − 12t + 5 = 0. Then t = (12 ± sqrt(144 − 98))/9.8 = (12 ± 6.78)/9.8, giving t = 0.53 s on the way up and t = 1.92 s on the way down.

Checking your answer

The two times should average to the time of the peak, which is u/g = 1.22 s. Here (0.53 + 1.92)/2 = 1.22 s, which matches.

Real-world uses

Estimating how long a ball stays in the air when it is thrown upward and caught at the same height.
Checking the timing of a bungee or jump drop.
A bouncing ball spends a predictable time in the air, which helps measure g in a classroom experiment with a stopwatch.

Open the interactive lesson, with simulation and quiz →

Frequently asked questions

Why is the time to go up the same as the time to come down?

Gravity is constant, so the deceleration on the way up mirrors the acceleration on the way down.

Does the time depend on mass?

No, not when air resistance is ignored.

Is the time to fall the same as the time to rise?

Yes, when air resistance is ignored. The motion is symmetric about the top, so the rise and fall take equal times.