section_path: Concepts derived from force/Kinematic integrals
section_name: Kinematic integrals
paragraph_text: Kinematic integrals
  
  Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. For example, integrating with respect to time gives the definition of impulse: J = ∫ t 1 t 2 F d t, {\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}{\mathbf {F} \,\mathrm {d} t},} which by Newton's Second Law must be equivalent to the change in momentum (yielding the Impulse momentum theorem).
  
  Similarly, integrating with respect to position gives a definition for the work done by a force: W = ∫ x 1 x 2 F ⋅ d x, {\displaystyle W=\int _{\mathbf {x} _{1}}^{\mathbf {x} _{2}}{\mathbf {F} \cdot {\mathrm {d} \mathbf {x} }},} which is equivalent to changes in kinetic energy (yielding the work energy theorem).
  
  Power P is the rate of change d W /d t of the work W, as the trajectory is extended by a position change d x {\displaystyle d\mathbf {x} } in a time interval d t: d W = d W d x ⋅ d x = F ⋅ d x, {\displaystyle \mathrm {d} W={\frac {\mathrm {d} W}{\mathrm {d} \mathbf {x} }}\cdot \mathrm {d} \mathbf {x} =\mathbf {F} \cdot \mathrm {d} \mathbf {x},} so P = d W d t = d W d x ⋅ d x d t = F ⋅ v, {\displaystyle P={\frac {\mathrm {d} W}{\mathrm {d} t}}={\frac {\mathrm {d} W}{\mathrm {d} \mathbf {x} }}\cdot {\frac {\mathrm {d} \mathbf {x} }{\mathrm {d} t}}=\mathbf {F} \cdot \mathbf {v},} with v = d x / d t {\displaystyle \mathbf {v} =\mathrm {d} \mathbf {x} /\mathrm {d} t} the velocity.