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Impulse (physics)

Impulse is the time integral of force that produces a change in momentum. It links force and motion, with applications from collisions to rocketry and safety design.

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Overview

In physics, impulse quantifies the effect of a force acting over a period of time. It is defined as the time integral of force: I = ∫ F dt and is equal to the change in linear momentum of a body. The quantity is a vector, often denoted I or J, and appears in classical mechanics formulations deriving from Newton's laws.

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Definition and formulae

The instantaneous mathematical definition is I = ∫ F(t) dt taken over the interval during which the force acts. For constant force, this reduces to I = F Δt. From Newton's second law (F = dp/dt) one obtains the impulse–momentum relation I = Δp, where p is momentum (p = m v for a constant mass). Units are newton-seconds (N·s) which are dimensionally equivalent to kg·m/s.

Behavior and examples

Impulse captures situations with large forces applied briefly (e.g., a hammer strike) as well as small forces over long times (e.g., a slowly accelerating rocket). In collisions, impulses determine post-impact velocities via conservation of momentum when external impulses are negligible. Automotive airbags and crumple zones increase the time over which force acts to reduce peak force while delivering the same impulse, improving occupant safety. See also the role of time in impulse calculations: time.

History and theoretical context

The impulse concept follows directly from Newtonian mechanics and the differential form F = dp/dt; integrating both sides over time yields the impulse–momentum theorem. Historically, impulse provided a practical way to connect measurable forces and durations with resulting changes in motion. Its mathematical representation uses the integral operator and links to the broader idea of momentum: momentum.

Properties and notable distinctions

  • Vector quantity: direction matches that of the net force.
  • Conservation context: in an isolated system, internal impulses cancel and total momentum is conserved.
  • Impulse vs instantaneous force: an idealized "impulse" can be modeled as an infinite force over infinitesimal time (impulse approximation), useful in collisions and rigid-body models.
  • Rotational analogue: torque integrated over time produces change in angular momentum.

Applications and measurement

Engineers and scientists compute impulse when designing protective equipment, analyzing sports impacts, sizing rocket burns (total impulse relates to propellant use), and interpreting crash data. Practically, impulse can be measured by force sensors and time sampling or by observing pre- and post-event velocities to compute Δp. For detailed mathematical methods see the integral treatment of forces and impulses: integral.

Definition, relations with mass and energy

Classical mechanics

The concept of momentum was introduced by Isaac Newton: He writes in Principia Mathematica:

"Quantitas motus est mensura ejusdem orta ex velocitate et quantitate materiae conjunctim."

"The magnitude of motion is measured by velocity and the magnitude of matter united."

By "magnitude of matter" is meant mass, and by "magnitude of motion" is meant momentum. Expressed in today's formula language, this definition thus reads:

{\vec p}=m\cdot {\vec v}

Since the mass ma scalar quantity, momentum {\vec {p}}and velocity are {\vec {v}}vectors with the same direction. Their magnitudes cannot be compared because they have different physical dimensions.

To change the velocity of a body (by direction and/or magnitude), its momentum must be changed. The momentum transferred divided by the time required to do so is the force {\vec {F}}:

{\displaystyle {\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\vec {F}}}

From the relationship between the momentum of a body and the force applied to it, the acceleration work performed is also related to the momentum:

{\displaystyle W=\int \limits _{C}{\vec {F}}({\vec {s}})\cdot \mathrm {d} {\vec {s}}=\int \limits _{C}{\vec {F}}({\vec {s}})\,\mathrm {d} t\cdot {\frac {\mathrm {d} {\vec {s}}}{\mathrm {d} t}}=\int \limits _{C}\mathrm {d} {\vec {p}}\cdot {\vec {v}}={\frac {1}{m}}\int \limits _{C}\mathrm {d} {\vec {p}}\cdot {\vec {p}}}

This work of acceleration is the kineticenergy. It follows

{\displaystyle E_{\text{kinetisch}}={\frac {{\vec {p}}^{\,2}}{2\,m}}={\frac {m\;{\vec {v}}^{\,2}}{2}}}.

Special theory of relativity

According to relativity, the momentum of a vbody moving with velocity with mass is given bym>0

{\vec p}={\frac {m\cdot {\vec v}}{{\sqrt {1-{v^{2} \over c^{2}}}}}}

given. In it cthe speed of light and always v<c. The momentum depends on the velocity nonlinearly, increasing towards infinity as the velocity of light is approached.

Generally valid is the energy-momentum-relation

{\displaystyle E^{2}-p^{2}\cdot c^{2}=m^{2}\cdot c^{4}.}

For objects with mass follows:

E = \frac{m \cdot c^2}{\sqrt{1-{v^2 \over c^2}}}

For v=0it follows p=0and E=m\,c^{2}(rest energy).

Objects without mass always move at the speed of light. For these it follows from the energy-momentum relation

E=p\,c

and this gives them the impulse

{\displaystyle p={E \over c}.}

Electromagnetic field

See also: Energy-momentum tensor

An electromagnetic field with electric field strength {\vec {E}}and magnetic field strength {\vec {H}}has the energy density

{\displaystyle u={\frac {1}{2}}\varepsilon _{0}E^{2}+{\frac {1}{2}}\mu _{0}H^{2}.}

These include the energy flux density (Poynting vector)

{\displaystyle {\vec {S}}={\vec {E}}\times {\vec {H}}}

and the pulse density

{\displaystyle \qquad {\vec {g}}={\frac {1}{c^{2}}}{\vec {E}}\times {\vec {H}}.}

Integrated over a given volume, these three expressions yield the energy E, the energy flux and the momentum p, which are associated with the total field present in that volume. For progressing plane waves again E=p\,c.

Conservation of momentum

→ Main article: Conservation of momentum

In an inertial system, momentum is a quantity of conservation. In a physical system on which no external forces act (in this context also called a closed system), the sum of all impulses of the components belonging to the system remains constant.

The initial total momentum is then also equal to the vector sum of the individual momentums present at any later time. Collisions and other processes within the system, in which the velocities of the components change, always end in such a way that this principle is not violated (see kinematics (particle processes)).

The conservation of momentum also applies to inelastic collisions. Although the kinetic energy decreases due to plastic deformation or other processes, the law of conservation of momentum is independent of the law of conservation of energy and applies to both elastic and inelastic collisions.

Impulse in pool: The impulse of the white ball is distributed to all balls.
Impulse in pool: The impulse of the white ball is distributed to all balls.

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