## Kinematic integrals::Force

### ::concepts

Force::forces    Object::first    Title::physics    Which::newton's    Motion::center    Velocity::constant

Kinematic integrals {{#invoke:main|main}} 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:<ref>{{#invoke:citation/CS1|citation |CitationClass=book }}</ref>

$\vec{I}=\int_{t_1}^{t_2}{\vec{F} \mathrm{d}t},$

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:<ref name=FeynmanVol1/>:13-3

$W=\int_{\vec{x}_1}^{\vec{x}_2}{\vec{F} \cdot{\mathrm{d}\vec{x}}},$

which is equivalent to changes in kinetic energy (yielding the work energy theorem).<ref name=FeynmanVol1/>:13-3

Power P is the rate of change dW/dt of the work W, as the trajectory is extended by a position change $\scriptstyle {d}\vec{x}$ in a time interval dt:<ref name=FeynmanVol1/>:13-2

$ \text{d}W\, =\, \frac{\text{d}W}{\text{d}\vec{x}}\, \cdot\, \text{d}\vec{x}\, =\, \vec{F}\, \cdot\, \text{d}\vec{x}, \qquad \text{ so } \quad P\, =\, \frac{\text{d}W}{\text{d}t}\, =\, \frac{\text{d}W}{\text{d}\vec{x}}\, \cdot\, \frac{\text{d}\vec{x}}{\text{d}t}\, =\, \vec{F}\, \cdot\, \vec{v}, $

with ${\vec{v}\text{ }=\text{ d}\vec{x}/\text{d}t}$ the velocity.

Force sections
Intro  Development of the concept  Pre-Newtonian concepts  Newtonian mechanics  Special theory of relativity  Descriptions  Fundamental forces  Non-fundamental forces  Rotations and torque  Kinematic integrals  Potential energy  Units of measurement  Force measurement  See also  Notes  References  Further reading  External links

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