In this article, we will just ignore air resistance for the sake of simplicity. Projectile motion happens when you throw an object at a certain angle, and the object (we called the object "projectile") moves along a curved path under the affect of gravitational force. If the projectile is launched with an initial velocity V₀ then in terms of vector it can be written as:

V₀=V_0xi + V_0yj

The components V_0x and V_0y can be found if we know the angle θ₀ between V₀ and the positive x direction:

V_0x=V₀ cos θ₀

V_0y=V₀ sin θ₀

On the direction of x-axis (horizonal direction), there's no acceleration, so a_x=0; meanwhile on vertical direction (y-axis) the acceleration a_y is always -g, because gravitational force is pointing downwards.

Since we ignore air resistance, we can say the velocity of horizonal component of the motion V_x is always V₀ cos θ₀, which equals to V_0x. For vertical direction, V_y will equal to  V_0y - gt, and that is equivalent to V₀ sin θ₀ - gt.