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[[Image:Brewsters-angle.svg|thumb|250px|An illustration of the polarization of light that is incident on an interface at Brewster's angle.]]
{{other uses}}
'''Brewster's angle''' (also known as the '''polarization angle''') is an [[angle of incidence]] at which [[light]] with a particular [[Polarization (waves)|polarization]] is perfectly transmitted through a transparent [[dielectric]] surface, with no [[Reflection (physics)|reflection]]. When ''unpolarized'' light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist [[David Brewster|Sir David Brewster]] (1781–1868).<ref>David Brewster (1815) [http://books.google.com/books?id=U-U_AAAAYAAJ&pg=PA125#v=onepage&q&f=false "On the laws which regulate the polarisation of light by reflection from transparent bodies,"] ''Philosophical Transactions of the Royal Society of London'', '''105''': 125-159.</ref>
{{redirect|Jostle|the racehorse|Jostle (horse)}}
A '''collision''' is an isolated event in which two or more moving bodies (colliding bodies) exert forces on each other for a relatively short time.


==Explanation==
A collision is not constrained to only referring to moving bodies it can also refer to electronic transactions which share a common resource such as a bus interface. In this case a collision refers to two simultaneous requests for the shared resource being made.
When light encounters a boundary between two [[medium (optics)|media]] with different [[refractive index|refractive indices]], some of it is usually reflected as shown in the figure above. The fraction that is reflected is described by the [[Fresnel equations]], and is dependent upon the incoming light's polarization and angle of incidence.  


The Fresnel equations predict that light with the ''p'' polarization ([[electric field]] polarized in the same [[Plane (mathematics)|plane]] as the [[incident ray]] and the [[surface normal]]) will not be reflected if the angle of incidence is
Although the most common colloquial use of the word "collision" refers to [[accident]]s in which two or more objects collide, the scientific use of the word "collision" implies nothing about the magnitude of the forces.


:<math>\theta_\mathrm B = \arctan \left( \frac{n_2}{n_1} \right), </math>
Some examples of physical interactions that scientists would consider collisions:  
* An insect touches its antenna to the leaf of a plant.  The antenna is said to collide with leaf.
* A cat walks delicately through the grass.  Each contact that its paws make with the ground is a collision.  Each brush of its fur against a blade of grass is a collision.


where ''n''<sub>1</sub> is the [[refractive index]] of the initial medium through which the light propagates (the "incident medium"), and ''n''<sub>2</sub> is the index of the other medium. This equation is known as '''Brewster's law''', and the angle defined by it is Brewster's angle.
Some colloquial uses of the word collision are:
* [[automobile collision]], two cars colliding
* [[mid-air collision]], two planes colliding
* [[ship collision]], two ships colliding


The physical mechanism for this can be qualitatively understood from the manner in which electric [[dipole]]s in the media respond to ''p''-polarized light. One can imagine that light incident on the surface is absorbed, and then reradiated by oscillating electric dipoles at the interface between the two media. The polarization of freely propagating light is always perpendicular to the direction in which the light is travelling. The dipoles that produce the transmitted (refracted) light oscillate in the polarization direction of that light. These same oscillating dipoles also generate the reflected light. However, dipoles do not radiate any energy in the direction of the [[Electric dipole moment|dipole moment]]. If the refracted light is p-polarized and propagates exactly perpendicular to the direction in which the light is predicted to be [[specular reflection|specularly reflected]], the dipoles point along the specular reflection direction and therefore no light can be reflected. (See diagram, above)
==Overview==
[[File:Deflection.png|right|thumb|250px|[[Deflection (physics)|Deflection]] happens when an object hits a plane surface.  If the kinetic energy after impact is the same as before impact, it is an elastic collision. If kinetic energy is lost, it is an inelastic collision. It is not possible to determine from the diagram whether the illustrated collision was elastic or inelastic, because no velocities are provided. The most one can say is that the collision was not perfectly-inelastic, because in that case the ball would have stuck to the wall.]]
Collision is short duration interaction between two bodies or more than two bodies simultaneously causeng change in motion of bodies involved due to internal forces acted between them during this . Collisions involve forces (there is a change in [[velocity]]). The magnitude of the velocity difference at impact is called the closing speed.  All collisions conserve [[momentum]].  What distinguishes different types of collisions is whether they also conserve [[kinetic energy]].Line of impact - It is the line which is common normal for surfaces are closest or in contact during impact. This is the line along which internal force of collision acts during impact and Newton's coefficient of restitution is defined only along this line.


With simple geometry this condition can be expressed as
Specifically, collisions can either be ''[[elastic collision|elastic]],'' meaning they conserve both momentum and kinetic energy, or ''[[inelastic collision|inelastic]],'' meaning they conserve momentum but not kinetic energy.  An inelastic collision is sometimes also called a ''plastic collision.''
:<math> \theta_1 + \theta_2 = 90^\circ,</math>
where θ<sub>1</sub> is the angle of reflection (or incidence) and θ<sub>2</sub> is the angle of refraction.


Using [[Snell's law]],
A “perfectly-inelastic” collision (also called a "perfectly-plastic" collision) is a limiting case of inelastic collision in which the two bodies stick together after impact.


:<math>n_1 \sin \left( \theta_1 \right) =n_2 \sin \left( \theta_2 \right),</math>
The degree to which a collision is elastic or inelastic is quantified by the [[coefficient of restitution]], a value that generally ranges between zero and one. A perfectly elastic collision has a coefficient of restitution of one; a perfectly-inelastic collision has a coefficient of restitution of zero.


one can calculate the incident angle θ<sub>1</sub>&nbsp;=&nbsp;θ<sub>B</sub> at which no light is reflected:
==Types of collisions==
There are two types of collision between two bodies- 1)head on collision or one dimensional collision - where nelocity of each just before impact is not alon line of impact. 2)non head on collision or oblique collision or two dimensional collision- where velocity of each is not long line of impact just before collision.


:<math>n_1 \sin \left( \theta_\mathrm B \right) =n_2 \sin \left( 90^\circ - \theta_\mathrm B \right)=n_2 \cos \left( \theta_\mathrm B \right).</math>
Though there are two special case of any collision as written below according to coefficient of restitution :


Solving for θ<sub>B</sub> gives
1)A perfectly [[elastic collision]] is defined as one in which there is no loss of [[kinetic energy]] in the collision.  In reality, any macroscopic collision between objects will convert some kinetic energy to [[internal energy]] and other forms of energy, so no large scale impacts are perfectly elastic.  However, some problems are sufficiently close to perfectly elastic that they can be approximated as such.Here coefficient of restitution is one.


:<math>\theta_\mathrm B = \arctan \left( \frac{n_2}{n_1} \right) .</math>
2)An [[inelastic collision]] is one in which part of the kinetic energy is changed to some other form of energy in the collision. [[Momentum]] is conserved in inelastic collisions (as it is for elastic collisions), but one cannot track the kinetic energy through the collision since some of it is converted to other forms of energy.Here coefficient of restitution is not one.
In any type of collision there is a phase when for a moment colliding bodies have same velocity along line of impact then kinetic energy of bodies reduces to its minimum during this phase and may be called as maximum deformation phase for which momentarily coefficient of restitution become one.
Collisions in [[ideal gases]] approach perfectly elastic collisions, as do scattering interactions of [[sub-atomic particles]] which are deflected by the [[electromagnetic force]]. Some large-scale interactions like the slingshot type gravitational interactions between satellites and planets are perfectly elastic.


For a glass medium (''n''<sub>2</sub>&nbsp;≈&nbsp;1.5) in air (''n''<sub>1</sub>&nbsp;≈&nbsp;1), Brewster's angle for visible light is approximately 56°, while for an air-water interface (''n''<sub>2</sub>&nbsp;≈&nbsp;1.33), it is approximately 53°. Since the refractive index for a given medium changes depending on the wavelength of light, Brewster's angle will also vary with wavelength.
Collisions between hard spheres may be nearly elastic, so it is useful to calculate the limiting case of an elastic collision. The assumption of conservation of momentum as well as the conservation of kinetic energy makes possible the calculation of the final velocities in two-body collisions.


The phenomenon of light being polarized by reflection from a surface at a particular angle was first observed by [[Étienne-Louis Malus]] in 1808.<ref>See:
==Analytical vs. numerical approaches towards resolving collisions ==
* Malus (1809) [http://books.google.com/books?id=hnJKAAAAYAAJ&pg=PA143#v=onepage&q&f=false "Sur une propriété de la lumière réfléchie"] (On a property of reflected light), ''Mémoires de physique et de chimie de la Société d'Arcueil'', '''2''' :  143-158.
Relatively few problems involving collisions can be solved analytically; the remainder require [[numerical methods]]. An important problem in simulating collisions is determining whether two objects have in fact collided.  This problem is called [[collision detection]].  
*  Malus, E.L. (1809) "Sur une propriété de la lumière réfléchie par les corps diaphanes" (On a property of light reflected by translucent substances), ''Nouveau Bulletin des Sciences'' [par la Societé Philomatique de Paris], '''1''' : 266-270.
*  Etienne Louis Malus, ''Théorie de la double réfraction de la lumière dans les substances christallisées'' [Theory of the double refraction of light in crystallized substances] (Paris, France:  Garnery, 1810).</ref> He attempted to relate the polarizing angle to the refractive index of the material, but was frustrated by the inconsistent quality of glasses available at that time. In 1815, Brewster experimented with higher-quality materials and showed that this angle was a function of the refractive index, defining Brewster's law.  


Brewster's angle is often referred to as the "polarizing angle", because light that reflects from a surface at this angle is entirely polarized perpendicular to the incident plane ("''s''-polarized") A glass plate or a stack of plates placed at Brewster's angle in a light beam can, thus, be used as a [[polarizer]]. The concept of a polarizing angle can be extended to the concept of a Brewster wavenumber to cover planar interfaces between two linear [[bianisotropic material]]s. In the case of reflection at Brewster's angle, the reflected and refracted rays are mutually perpendicular.
{{cleanup|section|date=February 2011}}.


== Applications ==
== Examples of collisions that can be solved analytically ==
===Billiards===
{{Anchor|Cue sports}}Collisions play an important role in [[cue sports]].  Because the collisions between [[billiard balls]] are nearly elastic, and the balls roll on a surface that produces low [[rolling friction]], their behavior is often used to illustrate [[Newton's laws of motion]]. After a zero-friction collision of a moving ball with a stationary one of equal mass, the angle between the directions of the two balls is 90 degrees. This is an important fact that professional billiards players take into account,<ref>{{cite web|last=Alciatore |first=David G. |date=January 2006 |url=http://billiards.colostate.edu/technical_proofs/TP_3-1.pdf |title=TP 3.1 90° rule |format=PDF |accessdate=2008-03-08 }}</ref> although it assumes the ball is moving frictionlessly across the table rather than rolling with friction.
Consider an elastic collision in 2 dimensions of any 2 masses m<sub>1</sub> and m<sub>2</sub>, with respective initial velocities '''u<sub>1</sub>'''  and '''u<sub>2</sub>''' = '''0''', and final velocities '''V<sub>1</sub>''' and '''V<sub>2</sub>'''.
Conservation of momentum gives m<sub>1</sub>'''u'''<sub>1</sub> = m<sub>1'''</sub>V<sub>1</sub>'''+ m<sub>2'''</sub>V<sub>2'''</sub>.
Conservation of energy for an elastic collision gives (1/2)m<sub>1</sub>|'''u<sub>1</sub>'''|<sup>2</sup> = (1/2)m<sub>1</sub>|'''V<sub>1</sub>'''|<sup>2</sup> + (1/2)m<sub>2</sub>|'''V<sub>2</sub>'''|<sup>2</sup>.
Now consider the case m<sub>1</sub> = m<sub>2</sub>: we obtain '''u<sub>1</sub>'''='''V<sub>1</sub>'''+'''V<sub>2</sub>''' and |'''u<sub>1</sub>'''|<sup>2</sup> = |'''V<sub>1</sub>'''|<sup>2</sup>+|'''V<sub>2</sub>'''|<sup>2</sup>.
Taking the [[dot product]] of each side of the former equation with itself, |'''u<sub>1</sub>'''|<sup>2</sup> = '''u<sub>1</sub>•u<sub>1</sub>''' = |'''V<sub>1</sub>'''|<sup>2</sup>+|'''V<sub>2</sub>'''|<sup>2</sup>+2'''V<sub>1</sub>•V<sub>2</sub>'''. Comparing this with the latter equation gives '''V<sub>1</sub>•V<sub>2</sub>''' = 0, so they are perpendicular unless '''V<sub>1</sub>''' is the zero vector (which occurs if and only if the collision is head-on).


Polarized [[sunglasses]] use the principle of Brewster's angle to reduce glare from the sun reflecting off horizontal surfaces such as water or road. In a large range of angles around Brewster's angle, the reflection of ''p''-polarized light is lower than ''s''-polarized light. Thus, if the sun is low in the sky, reflected light is mostly ''s''-polarized. Polarizing sunglasses use a polarizing material such as [[Polaroid (polarizer)|Polaroid]] sheets to block horizontally-polarized light, preferentially blocking reflections from horizontal surfaces. The effect is strongest with smooth surfaces such as water, but reflections from roads and the ground are also reduced.
===Perfectly inelastic collision===
[[Image:Inelastischer stoß.gif|a completely inelastic collision between equal masses]]


Photographers use the same principle to remove reflections from water so that they can photograph objects beneath the surface. In this case, the [[Polarizing filter (photography)|polarizing filter]] camera attachment can be rotated to be at the correct angle (see figure).
In a perfectly inelastic collision, i.e., a zero [[coefficient of restitution]], the colliding particles stick together. It is necessary to consider conservation of momentum:
::<math>m_a \mathbf u_a + m_b \mathbf u_b = \left( m_a + m_b \right) \mathbf v \,</math>
where '''v''' is the final velocity, which is hence given by
::<math>\mathbf v=\frac{m_a \mathbf u_a + m_b \mathbf u_b}{m_a + m_b}</math>
The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a [[center of momentum frame]] with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. In another frame, in addition to the reduction of kinetic energy there may be a transfer of kinetic energy from one particle to the other; the fact that this depends on the frame shows how relative this is.
With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a [[projectile]], or a [[rocket]] applying [[thrust]] (compare the [[Tsiolkovsky rocket equation#Derivation|derivation of the Tsiolkovsky rocket equation]]).


[[Image:Poloriser-demo.jpg|center|frame|Photograph taken of a window with a camera polarizer filter rotated to two different angles. In the picture at left, the polarizer is aligned with the polarization angle of the window reflection. In the picture at right, the polarizer has been rotated 90° eliminating the heavily polarized reflected sunlight.]]
== Examples of collisions analyzed numerically==
===Animal locomotion===
Collisions of an animal's foot or paw with the underlying substrate are generally termed ground reaction forces. These collisions are inelastic, as kinetic energy is not conserved. An important research topic in [[prosthetics]] is quantifying the forces generated during the foot-ground collisions associated with both disabled and non-disabled gait. This quantification typically requires subjects to walk across a [[force platform]] (sometimes called a "force plate") as well as detailed [[kinematic]] and [[Dynamics (mechanics)|dynamic]] (sometimes termed kinetic) analysis.


===Brewster windows=== <!--Brewster window redirects here.-->
== Collisions used as a experimental tool ==
[[Image:Brewster window.svg|thumb|right|A Brewster window]]
Collisions can be used as an experimental technique to study material properties of objects and other physical phenomena.
[[Gas laser]]s typically use a window tilted at Brewster's angle to allow the beam to leave the laser tube. Since the window reflects some ''s''-polarized light but no ''p''-polarized light, the round trip loss for the ''s'' polarization is higher than that of the ''p'' polarization. This causes the laser's output to be ''p'' polarized due to competition between the two modes.<ref name=Hecht>''Optics'', 3rd edition, Hecht, ISBN 0-201-30425-2</ref>


== See also ==
=== Space exploration===
* [[Brewster angle microscope]]
An object may deliberately be made to crash-land on another celestial body, to do measurements and send them to Earth before being destroyed, or to allow instruments elsewhere to observe the effect. See e.g.:
*During [[Apollo 13]], [[Apollo 14]], [[Apollo 15]], [[Apollo 16]] and [[Apollo 17]], the [[S-IVB]] (the rocket's third stage) was crashed into the [[Moon]] in order to perform seismic measurement used for characterizing the lunar core.
* [[Deep Impact (spacecraft)|''Deep Impact'']]
* [[SMART-1]] - [[European Space Agency]] satellite
* [[Moon impact probe]] - [[ISRO]] probe
 
=== Mathematical description of molecular collisions ===
Let the linear, angular and internal momenta of a molecule be given by the set of ''r'' variables { ''p''<sub>i</sub> }. The state of a molecule may then be described by the range ''δw''<sub>i</sub> = δ''p''<sub>1</sub>δ''p''<sub>2</sub>δ''p''<sub>3</sub> ... δ''p''<sub>r</sub>. There are many such ranges corresponding to different states; a specific state may be denoted by the index ''i''. Two molecules undergoing a collision can thus be denoted by (''i'', ''j'') (Such an ordered pair is sometimes known as a ''constellation''.)
It is convenient to suppose that two molecules exert a negligible effect on each other unless their centre of gravities approach within a critical distance ''b''. A collision therefore begins when the respective centres of gravity arrive at this critical distance, and is completed when they again reach this critical distance on their way apart. Under this model, a collision is completely described by the matrix <math>\begin{pmatrix}i&j\\k&l\end{pmatrix} </math>, which refers to the constellation (''i'', ''j'') before the collision, and the (in general different) constellation (''k'', ''l'') after the collision.
This notation is convenient in proving Boltzmann's [[H-theorem]] of [[statistical mechanics]].
 
==Attack by means of a deliberate collision==
Types of attack by means of a deliberate collision include:
* with the body: unarmed [[Strike (attack)|striking]], [[Punch (strike)|punching]], [[kick]]ing, [[martial arts]], [[pugilism]]
* striking directly with a weapon, such as a [[sword]], [[club (weapon)|club]] or [[axe]]
* [[ramming]] with an object or vehicle, e.g.:
** a car deliberately crashing into a building to break into it
** a [[battering ram]], medieval weapon used for breaking down large doors, also a modern version is used by police forces during raids
An attacking collision with a distant object can be achieved by throwing or launching a [[projectile]].
 
==See also==
{{multicol}}
*[[Ballistic pendulum]]
*[[Coefficient of restitution]]
*[[Collision detection]]
*[[Collision (telecommunications)]]
*[[Car accident]]
*[[Elastic collision]]
*[[Friction]]
* [[Head-on collision]]
*[[Impact crater]]
{{multicol-break}}
*[[Impact event]]
*[[Inelastic collision]]
*[[Kinetic theory]]<br /> - collisions between [[molecule]]s
*[[Mid-air collision]]
*[[Projectile]]
*[[Space debris]]
* [[Train wreck]]
{{multicol-end}}


==Notes==
==Notes==
<references/>
{{reflist|1}}


==References==
==References==
*A. Lakhtakia, "Would Brewster recognize today's Brewster angle?" ''OSA Optics News'', Vol. 15, No. 6, pp.&nbsp;14–18 (1989).
* {{cite book | author=Tolman, R. C. | title=The Principles of Statistical Mechanics | publisher=Clarendon Press | year=1938 | location=Oxford}} Reissued (1979) New York: Dover ISBN 0-486-63896-0.
*A. Lakhtakia, "General schema for the Brewster conditions," ''Optik'', Vol. 90, pp.&nbsp;184–186 (1992).


==External links==
==External links==
*[http://scienceworld.wolfram.com/physics/BrewstersAngle.html Brewster's Angle Extraction] from Wolfram Research
*[http://www.regispetit.com/bil_praa.htm Three Dimensional Collision] - Oblique inelastic collision between two homogeneous spheres.
*[http://www.rp-photonics.com/brewster_windows.html Brewster window at RP-photonics.com]
*[http://www.publicliterature.org/tools/collisions/ Two Dimensional Collision] - Java applet that simulates elastic collisions.
*[http://www.physics-lab.net/applets/one-dimensional-collisions One Dimensional Collision] - One Dimensional Collision Flash Applet.
*[http://www.physics-lab.net/applets/two-dimensional-collisions Two Dimensional Collision] - Two Dimensional Collision Flash Applet.
 
[[Category:Mechanics]]
[[Category:Introductory physics]]


{{DEFAULTSORT:Brewster's Angle}}
[[ar:تصادم]]
[[Category:Geometrical optics]]
[[cs:Ráz těles]]
[[Category:Physical optics]]
[[da:Kollision]]
[[Category:Angle]]
[[de:Stoß (Physik)]]
[[Category:Polarization (waves)]]
[[fa:برخورد]]
[[fr:Collision]]
[[ko:충돌]]
[[it:Urto]]
[[lt:Smūgis]]
[[hu:Ütközés]]
[[nl:Botsing (natuurkunde)]]
[[ja:衝突]]
[[pl:Zderzenie]]
[[pt:Colisão]]
[[ro:Coliziune]]
[[ru:Удар]]
[[simple:Collision]]
[[sl:Trk]]
[[sv:Stöt (mekanik)]]
[[zh:碰撞]]

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my page - www.hostgator1centcoupon.info A collision is an isolated event in which two or more moving bodies (colliding bodies) exert forces on each other for a relatively short time.

A collision is not constrained to only referring to moving bodies it can also refer to electronic transactions which share a common resource such as a bus interface. In this case a collision refers to two simultaneous requests for the shared resource being made.

Although the most common colloquial use of the word "collision" refers to accidents in which two or more objects collide, the scientific use of the word "collision" implies nothing about the magnitude of the forces.

Some examples of physical interactions that scientists would consider collisions:

  • An insect touches its antenna to the leaf of a plant. The antenna is said to collide with leaf.
  • A cat walks delicately through the grass. Each contact that its paws make with the ground is a collision. Each brush of its fur against a blade of grass is a collision.

Some colloquial uses of the word collision are:

Overview

Deflection happens when an object hits a plane surface. If the kinetic energy after impact is the same as before impact, it is an elastic collision. If kinetic energy is lost, it is an inelastic collision. It is not possible to determine from the diagram whether the illustrated collision was elastic or inelastic, because no velocities are provided. The most one can say is that the collision was not perfectly-inelastic, because in that case the ball would have stuck to the wall.

Collision is short duration interaction between two bodies or more than two bodies simultaneously causeng change in motion of bodies involved due to internal forces acted between them during this . Collisions involve forces (there is a change in velocity). The magnitude of the velocity difference at impact is called the closing speed. All collisions conserve momentum. What distinguishes different types of collisions is whether they also conserve kinetic energy.Line of impact - It is the line which is common normal for surfaces are closest or in contact during impact. This is the line along which internal force of collision acts during impact and Newton's coefficient of restitution is defined only along this line.

Specifically, collisions can either be elastic, meaning they conserve both momentum and kinetic energy, or inelastic, meaning they conserve momentum but not kinetic energy. An inelastic collision is sometimes also called a plastic collision.

A “perfectly-inelastic” collision (also called a "perfectly-plastic" collision) is a limiting case of inelastic collision in which the two bodies stick together after impact.

The degree to which a collision is elastic or inelastic is quantified by the coefficient of restitution, a value that generally ranges between zero and one. A perfectly elastic collision has a coefficient of restitution of one; a perfectly-inelastic collision has a coefficient of restitution of zero.

Types of collisions

There are two types of collision between two bodies- 1)head on collision or one dimensional collision - where nelocity of each just before impact is not alon line of impact. 2)non head on collision or oblique collision or two dimensional collision- where velocity of each is not long line of impact just before collision.

Though there are two special case of any collision as written below according to coefficient of restitution :

1)A perfectly elastic collision is defined as one in which there is no loss of kinetic energy in the collision. In reality, any macroscopic collision between objects will convert some kinetic energy to internal energy and other forms of energy, so no large scale impacts are perfectly elastic. However, some problems are sufficiently close to perfectly elastic that they can be approximated as such.Here coefficient of restitution is one.

2)An inelastic collision is one in which part of the kinetic energy is changed to some other form of energy in the collision. Momentum is conserved in inelastic collisions (as it is for elastic collisions), but one cannot track the kinetic energy through the collision since some of it is converted to other forms of energy.Here coefficient of restitution is not one. In any type of collision there is a phase when for a moment colliding bodies have same velocity along line of impact then kinetic energy of bodies reduces to its minimum during this phase and may be called as maximum deformation phase for which momentarily coefficient of restitution become one.

Collisions in ideal gases approach perfectly elastic collisions, as do scattering interactions of sub-atomic particles which are deflected by the electromagnetic force. Some large-scale interactions like the slingshot type gravitational interactions between satellites and planets are perfectly elastic.

Collisions between hard spheres may be nearly elastic, so it is useful to calculate the limiting case of an elastic collision. The assumption of conservation of momentum as well as the conservation of kinetic energy makes possible the calculation of the final velocities in two-body collisions.

Analytical vs. numerical approaches towards resolving collisions

Relatively few problems involving collisions can be solved analytically; the remainder require numerical methods. An important problem in simulating collisions is determining whether two objects have in fact collided. This problem is called collision detection.

Template:Cleanup.

Examples of collisions that can be solved analytically

Billiards

<Cue sports>...</Cue sports>Collisions play an important role in cue sports. Because the collisions between billiard balls are nearly elastic, and the balls roll on a surface that produces low rolling friction, their behavior is often used to illustrate Newton's laws of motion. After a zero-friction collision of a moving ball with a stationary one of equal mass, the angle between the directions of the two balls is 90 degrees. This is an important fact that professional billiards players take into account,[1] although it assumes the ball is moving frictionlessly across the table rather than rolling with friction. Consider an elastic collision in 2 dimensions of any 2 masses m1 and m2, with respective initial velocities u1 and u2 = 0, and final velocities V1 and V2. Conservation of momentum gives m1u1 = m1V1+ m2V2. Conservation of energy for an elastic collision gives (1/2)m1|u1|2 = (1/2)m1|V1|2 + (1/2)m2|V2|2. Now consider the case m1 = m2: we obtain u1=V1+V2 and |u1|2 = |V1|2+|V2|2. Taking the dot product of each side of the former equation with itself, |u1|2 = u1•u1 = |V1|2+|V2|2+2V1•V2. Comparing this with the latter equation gives V1•V2 = 0, so they are perpendicular unless V1 is the zero vector (which occurs if and only if the collision is head-on).

Perfectly inelastic collision

a completely inelastic collision between equal masses

In a perfectly inelastic collision, i.e., a zero coefficient of restitution, the colliding particles stick together. It is necessary to consider conservation of momentum:

where v is the final velocity, which is hence given by

The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. In another frame, in addition to the reduction of kinetic energy there may be a transfer of kinetic energy from one particle to the other; the fact that this depends on the frame shows how relative this is. With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a projectile, or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation).

Examples of collisions analyzed numerically

Animal locomotion

Collisions of an animal's foot or paw with the underlying substrate are generally termed ground reaction forces. These collisions are inelastic, as kinetic energy is not conserved. An important research topic in prosthetics is quantifying the forces generated during the foot-ground collisions associated with both disabled and non-disabled gait. This quantification typically requires subjects to walk across a force platform (sometimes called a "force plate") as well as detailed kinematic and dynamic (sometimes termed kinetic) analysis.

Collisions used as a experimental tool

Collisions can be used as an experimental technique to study material properties of objects and other physical phenomena.

Space exploration

An object may deliberately be made to crash-land on another celestial body, to do measurements and send them to Earth before being destroyed, or to allow instruments elsewhere to observe the effect. See e.g.:

Mathematical description of molecular collisions

Let the linear, angular and internal momenta of a molecule be given by the set of r variables { pi }. The state of a molecule may then be described by the range δwi = δp1δp2δp3 ... δpr. There are many such ranges corresponding to different states; a specific state may be denoted by the index i. Two molecules undergoing a collision can thus be denoted by (i, j) (Such an ordered pair is sometimes known as a constellation.) It is convenient to suppose that two molecules exert a negligible effect on each other unless their centre of gravities approach within a critical distance b. A collision therefore begins when the respective centres of gravity arrive at this critical distance, and is completed when they again reach this critical distance on their way apart. Under this model, a collision is completely described by the matrix , which refers to the constellation (i, j) before the collision, and the (in general different) constellation (k, l) after the collision. This notation is convenient in proving Boltzmann's H-theorem of statistical mechanics.

Attack by means of a deliberate collision

Types of attack by means of a deliberate collision include:

An attacking collision with a distant object can be achieved by throwing or launching a projectile.

See also

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Notes

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References

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    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Reissued (1979) New York: Dover ISBN 0-486-63896-0.

External links

ar:تصادم cs:Ráz těles da:Kollision de:Stoß (Physik) fa:برخورد fr:Collision ko:충돌 it:Urto lt:Smūgis hu:Ütközés nl:Botsing (natuurkunde) ja:衝突 pl:Zderzenie pt:Colisão ro:Coliziune ru:Удар simple:Collision sl:Trk sv:Stöt (mekanik) zh:碰撞

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