Projectile motion refers to the motion of an object that is launched into the air and moves along a curved path under the influence of gravity. It is a combination of horizontal and vertical motion. To understand projectile motion in detail, we can break down the derivation into several key points:

1. Initial velocity components:
- When an object is launched, it has an initial velocity that can be resolved into horizontal and vertical components.
- The horizontal component remains constant throughout the motion, while the vertical component changes due to the influence of gravity.
- Let's consider the initial velocity as V0, the horizontal component as V0x, and the vertical component as V0y.

2. Horizontal motion:
- In the absence of any external forces, the horizontal motion of a projectile remains constant.
- There is no acceleration in the horizontal direction, so the horizontal component of the velocity remains constant.
- The horizontal distance traveled by the projectile, also known as the range (R), can be calculated using the formula R = V0x * t, where t represents time.

3. Vertical motion:
- In the vertical direction, the projectile experiences a constant acceleration due to gravity, which is typically represented as g (9.8 m/s²).
- The initial vertical velocity (V0y) decreases as the object rises until it reaches its maximum height, and then increases as it falls.
- The time taken for the projectile to reach its maximum height (tmax) can be determined using the formula tmax = V0y / g.
- The maximum height (H) reached by the projectile can be calculated using the formula H = (V0y)² / (2 * g).
- The total time of flight (T) can be found by doubling the time taken to reach the maximum height, T = 2 * tmax.

4. Projectile motion equations:
- The horizontal position of the projectile (x) can be calculated using x = V0x * t.
- The vertical position of the projectile (y) can be determined using y = V0y * t - (1/2) * g * t².
- These equations allow us to calculate the position of the projectile at any given time.

In summary, projectile motion involves the motion of an object launched into the air, experiencing both horizontal and vertical components of motion. The horizontal motion remains constant, while the vertical motion is influenced by gravity. By analyzing the initial velocity components and applying the equations of motion, we can accurately predict the trajectory of a projectile.

When a particle is thrown obliquely near the earth surface it moves along a curved path. such particle is called projectile and its motion is called projectile motion. In vertical and horizontal motion are ; in horizontal motion are , Ax = 0 ; Vx = Ux + Axt ==> Ux = u cos thetha . ;; x = Uxt + 1/2axt^2 = uxt. and vertical motion , vy = uy - gt , y = uyt - 1/2 gt^2 , vy^2 = uy^2 - 2gy. ( formula comes from equation of motion )