section_path: Newtonian mechanics/Second law
section_name: Second law
paragraph_text: 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 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 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.
  
  A modern statement of Newton's second law is a vector equation: F = d p d t, {\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}},} where p {\displaystyle \mathbf {p} } is the momentum of the system, and F {\displaystyle \mathbf {F} } is the net (vector sum) force. 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.
  
  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, F = d p d t = d (m v) d t, {\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} \left(m\mathbf {v} \right)}{\mathrm {d} t}},} where m is the mass and v {\displaystyle \mathbf {v} } is the velocity. If Newton's second law is applied to a system of constant mass, m may be moved outside the derivative operator. The equation then becomes F = m d v d t. {\displaystyle \mathbf {F} =m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}.} By substituting the definition of acceleration, the algebraic version of Newton's second law is derived: F = m a. {\displaystyle \mathbf {F} =m\mathbf {a}.}