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=V0xi + V0yj
The components V0x and V0y can be found if we know the angle θ0 between V0 and the positive x direction:
V0x=V0 cos θ0
V0y=V0 sin θ0
On the direction of x-axis (horizonal direction), there's no acceleration, so ax=0; meanwhile on vertical direction (y-axis) the acceleration ay 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 Vx is always V0 cos θ0, which equals to V0x. For vertical direction, Vy will equal to V0y - gt, and that is equivalent to V0 sin θ0 - gt.