All our efforts led the way to the new inventions and discoveries. Gravity and Gravitational waves are one among the best discoveries ever made. We can find gravity in any part of the universe. Even in the space, we find Huge Black holes with high gravitational force which have a devastating effect. We are also aware of Worm Holes. So, we came to know that each and every particle on the earth has some gravitational force up to some height from the ground. A ball when thrown from the ground to some height in a straight path again reaches the ground in the same straight path. When the ball is thrown in a curved position ( not a straight path ), then it moves to some distance. This is known as projectile and this type of motion is known as projectile motion.

For a projectile we have two types of displacements. The Horizontal Displacement and the Vertical Displacement. The distance covered by the projectile in the horizontal direction is known as Horizontal Displacement and the distance covered by the projectile in the vertical direction is known as Vertical Displacement. There are even two types of projectiles.

- Horizontal Projectile
- NonHorizontal or Slant Projectile

**Horizontal Projectile** is the projectile that is launched with 0 degrees or 180 degrees with respect to the surface. This can be treated as the combination of vertical distance and horizontal distance only in the terms of components. As all the projectiles are based on the Laws of Motion and on the Work Energy Theorem, it must touch the ground at some distance.

**Non Horizontal projectile** is the projectile that is launched with more than 0 degree or less than 180 degrees with respect to the horizontal or surface. Even this touches the ground at some distance. Non Horizontal projectiles takes long time of journey when compared to the Horizontal projectiles.

The same as horizontal and vertical displacements, there also exists Horizontal and vertical velocities. At any instant of time, we can divide the point into horizontal velocity and vertical velocity. The horizontal velocity is represented by a horizontal line and the vertical velocity is represented by a vertical line. Let us say the angle of projectile from the point to the surface is ϴ, then the horizontal velocity is velocity multiplied by cos ϴ and the vertical velocity is the velocity multiplied by sin ϴ.

V is the velocity of the projectile.

ϴ is the angle between the.

The horizontal component of velocity is V * Cos ϴ.

The vertical component of velocity is V * Sin ϴ.

The Velocity also changes with respect to time and with respect to the position of the object.

For determining the components of Velocity Vector, we need to have initial velocity. In simple words, we need to have initial displacement and velocity components of both horizontal or vertical components. If this is not possible, we need to have initial velocity and the angle with which the object is projected. The perpendicular components- Vertical component and horizontal component are independent of each other and depend only on the initial boost up velocity and the angle projected in to the air.

We have initial velocity Vo = Vox*i + Voy*j

Vox = Vo * Cos ϴ

Voy = Vo * Sin ϴ

Kinematic Equations-

Acceleration in the downward direction is taken positive and the acceleration in upward direction is taken negative.

ax = 0

ay = -g

where g = 9.82 m/s^2

Vx = Vo Cos ϴ

Vy = Vo sin ϴ – gt

t = time in seconds.

V = ( Vx^2 + Vy^2 )^(1/2)

V= Magnitude of Velocity.

Displacement x = Vot Cos ϴ, y = Vot sin ϴ – gt^2/2

These are parabolic equations.

we know

y = Vot Sin ϴ – gt^2/2

T = 2Vo Sin ϴ/ g

T= Time of Flight of the total time of journey of the projectile from the time of projection to the dropping on to the ground.

Maximum height of the projectile-

H max = ( Vo * Sin ϴ )^2/ 2g

H max = Maximum height that the projectile can reach from the surface.

The maximum distance that the projectile can travel is at the angle of projection at 45 degrees.

The minimum distance that the projectile can travel is at the angle of projection at 90 degrees.

We have come to know about Kinematic equations of particle motion namely Acceleration, velocity, Displacement and time. All the equations including Range of projectile, Maximum height of the projectile, Maximum displacement of the projectile and time of flight are described with equations. Work Energy theorem will also be helpful in formulating these equations.

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