section_path: Newtonian mechanics/Work and energy
section_name: Work and energy
paragraph_text: Work and energy
  
  If a constant force F is applied to a particle that makes a displacement Δ r, the work done by the force is defined as the scalar product of the force and displacement vectors:
  
  More generally, if the force varies as a function of position as the particle moves from r to r along a path C, the work done on the particle is given by the line integral
  
  If the work done in moving the particle from r to r is the same no matter what path is taken, the force is said to be conservative. Gravity is a conservative force, as is the force due to an idealized spring, as given by Hooke's law. The force due to friction is non-conservative.
  
  The kinetic energy E of a particle of mass m travelling at speed v is given by
  
  For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.
  
  The work–energy theorem states that for a particle of constant mass m, the total work W done on the particle as it moves from position r to r is equal to the change in kinetic energy E of the particle:
  
  Conservative forces can be expressed as the gradient of a scalar function, known as the potential energy and denoted E:
  
  If all the forces acting on a particle are conservative, and E is the total potential energy (which is defined as a work of involved forces to rearrange mutual positions of bodies), obtained by summing the potential energies corresponding to each force
  
  The decrease in the potential energy is equal to the increase in the kinetic energy
  
  This result is known as conservation of energy and states that the total energy,
  
  is constant in time. It is often useful, because many commonly encountered forces are conservative.