Impulse (physics)

This article describes the physical quantity momentum, which characterizes the mechanical state of motion of a physical object. For other meanings, see Impulse (disambiguation).

Momentum is a fundamental physical quantity that characterizes the mechanical state of motion of a physical object. The momentum of a physical object is greater the faster it moves and the more massive it is. Thus, momentum stands for what in colloquial language is vaguely referred to as "momentum" and "momentum".

The formula symbol of the impulse is usually (from Latin pellere 'to push, to drive'). The unit is kg-m-s-1 = N-s in the International System of Units.

Unlike kinetic energy, momentum is a vector quantity and thus has a magnitude and a direction. Its direction is the direction of motion of the object. Its magnitude in classical mechanics is given by the product of the mass of the body and the velocity of its center of mass. In relativistic mechanics, a different formula (quadruple momentum) applies, which approximates the classical formula for velocities far below the speed of light. However, it also attributes momentum to massless objects moving at the speed of light, e.g. classical electromagnetic waves or photons.

The momentum of a body characterizes exclusively the translational motion of its center of mass. Any additional rotation about the center of mass is described by the angular momentum. The momentum is an additive quantity. The total momentum of an object with several components is the vector sum of the momentums of its parts.

The momentum, like the velocity and the kinetic energy, depends on the choice of the reference frame. In a fixed inertial frame, momentum is a quantity of conservation, i.e. an object on which no external forces act maintains its total momentum in terms of magnitude and direction. If two objects exert force on each other, for example in a collision, their two momentums change in opposite ways such that their vectorial sum is conserved. The amount by which the momentum changes for one of the objects is called the momentum transfer. In the context of classical mechanics, momentum transfer is independent of the choice of inertial frame.

The concept of momentum developed from the search for a measure of the "quantity of motion" present in a physical object, which, according to all experience, is maintained in all internal processes. This explains the now obsolete terms "magnitude of motion" or "quantity of motion" for the impulse. Originally, these terms could also refer to kinetic energy; it was not until the beginning of the 19th century that a clear distinction was made between the terms. In English, momentum is called momentum, while impulse refers to the transfer of momentum (force impact).

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 a 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 ${\vec {F}}$ :

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

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:

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

This work of acceleration is the kineticenergy. It follows

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

Special theory of relativity

According to relativity, the momentum of a body moving with velocity with mass is given by m>0

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

given. In it the 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

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

For objects with mass follows:

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

For v=0 it follows p=0 and $E=m\,c^{2}$ (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

$p={E \over c}.$

Electromagnetic field

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.