how to mathematically determine projectile motion?

according to Newton’s second law, F=ma

integrating the constant acceleration twice gives the equation at time t of x(t) = x(0) + vx(0)t + 0.5*ax*t^2. Here, ax is the constant acceleration in the direction of the x-axis, vx(0) is the initial velocity along the x-axis, and x(0) is the initial position.