# Trajectory

## What is Trajectory?

Trajectory is the curved path followed by an object with a certain mass while moving through space with respect to time. It is also known as Flight Path.

Projectile motion gives rise to trajectory as it is such a motion in which the moving object follows a curved path in the air due to gravity.

## Trajectory in Physics

We all know that whenever an object is thrown upwards, then, due to the gravitation pull of the earth, after reaching a certain height and travelling a specific distance, it comes back to the earth after some time.

So, the trajectory in physics simply tells about the path of the projectile in space due to gravity. More specifically, it is a parabolic path in the air.

Galileo first provided the idea of the curved path of an object in space.

Similarly, in terms of mathematics, trajectory describes the position of an object over a particular instant of time.

### Examples

The various examples of trajectory are as follows:

• The path of a bullet fired from a gun.
• The motion of a paper airplane in the air results in trajectory.
• The orbit around earth by a satellite.
• The path followed by the stone thrown in a river.

## Trajectory Formula

So far, we know that the parabolic path of the projectile as a function of time due to gravity is defined as Trajectory.

As trajectory possesses both horizontal and vertical positions hence, by the use of the trajectory formula, we can have the vertical position of the projectile with the other known values.

Suppose a ball is thrown upwards that undergoes projectile motion. Consider that the figure given below shows the motion of the ball in space: Here x and y represent the horizontal and vertical position of the object in meters, respectively, while v0 is the initial velocity with which the ball is thrown. Also, g represents gravity (= 9.8m/s2), and θ represents the angle of projection from the horizontal plane.

In trajectory, the path possessed by the ball depends on the gravitational force and resistance of air. Here, path OBD represents the trajectory of the projectile, and it is parabolic in nature.

So, the vertical position of the ball in space is given by the formula: Thus, for the above figure, we can write it as: Hence, by the use of the above-given formula, one can determine the vertical position of the ball in the space.

### Example

Let’s now understand how the vertical position of the projectile can be determined by using the above-given formula.

Suppose a person is standing at a riverbank and throws a stone at an initial velocity of 30 m/s at an angle of 60° in relation to the riverbank. Also, considering that the stone reaches the other side of the river, which is 10 m away. So, determine the height of the stone when it reaches the other end of the river bank from where the stone is thrown.

Here,

• the height corresponds to the vertical position = y,
• the horizontal position = distance between the river banks i.e., x = 10m,
• angle θ = 60° and
• initial velocity v0 = 30m/s

So, according to the formula of trajectory, On substituting the respective values, Hence, for the above given example, the vertical position of the stone reaching the other end is 15.14 m.