Time of Flight Formula

One often experiences many kinds of motions in their daily life. Projectile motion is one of them. A projectile is somebody thrown in space or air with some initial force and curved path. The Curved path through which the projectile travels is what is termed as its trajectory. For example, the free-fall motion of any object in a horizontal path with constant velocity is a type of Projectile Motion. This motion has many terms for computations such as horizontal velocity, vertical velocity, Maximum height, time of flight, etc. In this article, we will discuss the time of flight formula with examples. Let us begin learning!

Time of Flight Formula

What is Projectile Motion?

Projectile motion is a form of motion in which an object moves in a bilaterally symmetrical and parabolic path. The path traced by the object is called its trajectory. Projectile motion occurs only when there is some force applied at the beginning on the trajectory. After this initial thrust, the only interference is from gravity. In real life, many applications are found using projectile motion.

Many physical terms are computed in projectile motion. Some of these are as follows:

 1.Initial Velocity:

The initial velocity can be expressed as x components and y components:

\( u_\text{x} = \text{u} \cdot \cos\theta \)

\( u_\text{y} = \text{u} \cdot \sin\theta \)

where,

\( u_\text{x} \)	The horizontal component of velocity
\( u_\text{y} \)	The vertical component of velocity
u	Velocity magnitude
\( \theta \)	Projectile Angle

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 2.Time of Flight:

The time of flight of projectile motion is defined as the time from when the object is projected to the time it reaches the surface. As we discussed previously, TT depends on the initial velocity magnitude and the angle of the projectile:

\( T=\frac{2 \cdot \text{u}_\text{y}}{\text{g}} \)

i.e.\( T=\frac{2 \cdot \text{u} \cdot \sin\theta}{\text{g}} \)

where,

\( u_\text{x} \)	The horizontal component of velocity
\( u_\text{y} \)	The vertical component of velocity
T	Time of flight
u	Velocity magnitude
\( \theta \)	Projectile Angle
g	Acceleration due to gravity

3.Velocity:

The horizontal velocity remains constant, but the vertical velocity varies linearly because the acceleration is constant. At any time, t, the velocity is:

\( u_x = u \cdot \cos\theta \)

\( u_y = u \cdot \sin \theta – g\cdot t \)

You can also use the Pythagorean Theorem to find velocity:

\( u=\sqrt{u_x^2+u_y^2} \)

Where,

\(u_\text{x}\)	The horizontal component of velocity
\(u_\text{y} \)	The vertical component of velocity
T	Time of flight
U	Velocity magnitude
\(\theta \)	Projectile Angle

4. Maximum Height:

The maximum height is reached when v_y=0. Using this we can rearrange the velocity equation to find the time it will take for the object to reach maximum height.

\(h=\frac{u^2 \cdot \sin^2\theta}{2\cdot g} \)

where,

h	Maximum height
g	Acceleration due to gravity
u	Velocity magnitude
\(\theta \)	Projectile Angle

Solved Examples on Time of Flight Formula

Q. 1: A body is projected with a velocity of \(20 ms^{-1} \) at 50° to the horizontal plane. Find the time of flight of the projectile.

Solution:

Initial Velocity Vo = \(20 ms^{-1} \)

And angle \(\theta = 50° \)

So, Sin 50° =  0.766

And g= 9.8

Now formula for time of flight is,

T = \( \frac {2 \cdot \text{u} \cdot \sin\theta}{\text{g}} \)

T = \(\frac {2 \times 20 \times \sin 50°}{9.8}\)

= \( \frac {2\times 20 \times0.766}{9.8}\)

= \( \frac{30.64}{9.8}\)

T = 3.126 sec

Therefore time of flight is 3.126 second.