Force

=== Second law ===
{{main|Newton's second law}}
According to the first law, motion at constant speed in a straight line does not need a cause. It is ''change'' in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion. 
[[Newton's second law]] states that the net force acting upon an object is equal to the [[time derivative|rate]] at which its [[momentum]] changes with [[time]]. If the mass of the object is constant, this law implies that the [[acceleration]] of an object is directly [[Proportionality (mathematics)|proportional]] to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the [[mass]] of the object.<ref name="openstax-university-physics" />{{rp|pages=204–207}}

A modern statement of Newton's second law is a vector equation:
<math display="block" qid=Q104212301>\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t},</math>
where <math> \vec{p}</math> is the momentum of the system, and <math> \vec{F}</math> is the net ([[Vector (geometric)#Addition and subtraction|vector sum]]) force.<ref name="openstax-university-physics" />{{rp|page=399}} If a body is in equilibrium, there is zero ''net'' force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an ''unbalanced'' force acting on an object it will result in the object's momentum changing over time.<ref name="Principia"/>

In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum,
<math display="block">\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t} = \frac{\mathrm{d}\left(m\vec{v}\right)}{\mathrm{d}t},</math>
where ''m'' is the [[mass]] and <math> \vec{v}</math> is the [[velocity]].<ref name=FeynmanVol1/>{{rp|((9-1,9-2))}} If Newton's second law is applied to a system of [[Newton's Laws of Motion#Open systems|constant mass]], ''m'' may be moved outside the derivative operator. The equation then becomes
<math display="block">\vec{F} = m\frac{\mathrm{d}\vec{v}}{\mathrm{d}t}.</math>
By substituting the definition of [[acceleration]], the algebraic version of [[Newton's second law]] is derived:
<math display="block" qid=Q2397319>\vec{F} =m\vec{a}.</math>