What is Time of Flight?
In physics, **Time of Flight** ($T$) is the total amount of time an object, known as a projectile, spends in the air. This calculation is a cornerstone of kinematics, the study of motion, and is essential for analyzing any object launched, thrown, or shot. From calculating the arc of a basketball to determining the range of a cannon, understanding time of flight is critical.
The time an object spends in the air is determined purely by its vertical motion. Factors like initial vertical velocity, initial height, and the acceleration of gravity dictate how long it takes for the object to travel up to its peak (if applicable) and fall back to the ground.
The General Time of Flight Formula
To find the time of flight, we start with the kinematic equation for vertical position ($y$):
Where:
- $y_f$ is the final vertical position (which we set to 0 for landing on the ground).
- $y_i$ is the initial vertical position, or initial height ($h_0$).
- $v_{iy}$ is the initial vertical velocity.
- $a_y$ is the vertical acceleration, which is gravity ($-g$).
Substituting our variables, we get: $0 = h_0 + (v_0 \sin(\theta))t - \frac{1}{2}gt^2$. This is a quadratic equation. By solving for $t$ using the quadratic formula, we get the general equation for the time of flight:
The components are:
- $v_0 \sin(\theta)$: The initial vertical velocity ($v_{0y}$).
- $g$: The acceleration due to gravity (approx. 9.8 m/s²).
- $h_0$: The initial height from which the projectile is launched.
Special Case: Launching from the Ground
The most common scenario in introductory physics is when a projectile is launched from the ground and lands on the ground. In this case, the initial height ($h_0$) is 0.
When $h_0 = 0$, the general formula simplifies significantly:
$T = \frac{v_0 \sin(\theta) + \sqrt{(v_0 \sin(\theta))^2 + 2g(0)}}{g}$
$T = \frac{v_0 \sin(\theta) + \sqrt{(v_0 \sin(\theta))^2}}{g}$
$T = \frac{v_0 \sin(\theta) + v_0 \sin(\theta)}{g}$
This gives us the simple, well-known formula for time of flight from a flat surface:
In this symmetrical arc, the time to reach the peak height is $\frac{v_0 \sin(\theta)}{g}$, and the time to fall back down is the same, doubling the total time.
What About Horizontal Motion?
You may have noticed that the horizontal velocity ($v_x = v_0 \cos(\theta)$) is not in the time of flight formula. This is a critical concept: **horizontal motion does not affect vertical motion.**
An object dropped straight down from 100 meters (like in our Potential Energy Calculator) will hit the ground at the exact same time as an object shot horizontally from a cannon at 100 meters (ignoring air resistance).
However, you can use the Time of Flight ($T$) to find the total horizontal distance the projectile travels, known as its **Range ($R$)**. The formula for range is simple:
For more information on the principles of projectile motion, the Wikipedia article on Projectile Motion is an excellent resource.