Why the Newton's second law of motion important?

of Motion (##\vec F_\text{Net} =m\vec a##) is very important because it shows the relationship between forces and motion. It allows you to calculate the (and therefore velocity and position) of an object with known forces.

This is incredibly valuable for scientists, engineers, inventors, etc. By controlling, measuring, or calculating the force exerted on objects, we can map out their trajectory, thus knowing where they will be at any given time.

Note: Newton's Second Law, when expressed as ##\vec F_\text{net} = m\vec a##, is only valid at non-relativistic speeds, also it is only valid for cases of constant mass. In cases with variable mass,

##\vec F_{\text{net}} = \frac{d \vec p}{dt } = m \frac{\partial \vec v}{\partial t} + \vec v\frac{\partial m}{\partial t}## where ##\vec p## is the momentum, ##m## is the mass, and ##vec v## is the velocity. The final term above is called thrust, and is the primary acceleration mechanism behind rockets and jet engines.