**Trajectory Formula**

The trajectory formula is used to determine the position of an object at any given time in the future. It can be used for anything from finding the location of a planet to the path of a bullet fired from a gun to the movement of a baseball through the air. We can use it to find out how far away an object will be from its starting position at any given moment.

**About the Topic **

The trajectory can be defined in a few ways. It can be the distance that an object follows when thrown into the air. It can also be described as a line followed by a moving point. The word trajectory also refers to the path taken by a projectile as it moves through space, and how the projectile will change directions during its flight.

The word, trajectory, is mostly used with reference to balls being thrown or kicked in sports, where it is assumed that the ball will not rise or fall more than very slightly. However, the word is sometimes applied to the flight of anything that flies through the air.

**Formula**

y = yo + (vosinθ) – 1/2gt2

y = displacement

y0 = Initial velocity

v0 = final velocity

g = 9.8 m/s2

t = total time

**Solved Examples**

**If the initial velocity of a stone thrown by a boy is 6 m/sec, and the angle at which the stone is thrown is 60****∘****. Find the equation of the path of the projectile. Use g = 9.8 m/sec****2****. Solve this by using the trajectory formula.**

Given, θ = 60∘.

v(initial velocity) = 6m /sec

Using the trajectory formula,

y = yo + (vosinθ) – 1/2gt2

y = xtanθ − gx2/2v2cos2θ

y = x tan 60 – (9.8)(x2)/(2)(62)(cos2 60)

y = x√3 – 0.544x2

**If Trevor hits a ball with his bat at an initial velocity of 45 m/s in the air. In the ball’s direction of travel, the end of the field is 140.0 m away. If the initial angle at which the ball is thrown is 66.4°. Calculate the vertical height when the ball reaches the end of the field. Solve this by using the trajectory formula.**

Given, θ = 66.4°

v = 45 m/s

x = 140.0 m

Using the trajectory formula,

y = (140)(tan 66.4°) – [(9.8)(140)(140)/(2)(45)2(0.4)2]

y = 320.6 – 192080/648

y = 320.6 – 296.4 = 24.2