Thus,{\displaystyle \mathbf {F} =m\,{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=m\mathbf {a} ,}

where **F** is the net force applied, *m* is the mass of the body, and **a** is the body’s acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it. An application of this notation is the derivation of G Subscript C.

The above statements hint that the second law is merely a definition of {\displaystyle \mathbf {F} }, not a precious observation of nature. However, current physics restate the second law in measurable steps: (1)defining the term ‘one unit of mass’ by a specified stone, (2)defining the term ‘one unit of force’ by a specified spring with specified length, (3)measuring by experiment or proving by theory (with a principle that every direction of space are equivalent), that force can be added as a mathematical vector, (4)finally conclude that {\displaystyle \mathbf {F} =m\mathbf {a} }. These steps hint the second law is a precious feature of nature.

The second law also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the momentum.

Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems (see below).

Newton’s second law is an approximation that is increasingly worse at high speeds because of relativistic effects.

According to modern ideas of how Newton was using his terminology