利用者:紅い目の女の子/反発係数

{{物理量 | 名称 = | 英語 = coefficient of restitution | 画像 =[[ファイル:Colorful_Super_ball.jpg|thumb|250px]] | 記号 =''e'' | 次元 =''[[無次元数|無次元量]]'' | 階 =スカラー | SI = | CGS = | MTS = | FPS = | MKSG = | CGSG = | FPSG = | プランク = | 原子 = }} 反発係数（はんぱつけいすう、coefficient of restitution）は、2物体の衝突において、衝突前の互いに近づく速さに対する、衝突後の互いに遠ざかる速さの比のことである（衝突の前後での相対速度の大きさの比）。はねかえり係数ともいう。普通、文字e で示し、0≦e ≦1の範囲をとる単位がない値（無次元数）である。

衝突時に2物体の間でのみ力がはたらく場合、2物体全体の運動量の和は一定であるが、運動エネルギーの和は一定とは限らない。一般に衝突時には音や温度上昇が生じるので、運動エネルギーの一部が他の形態のエネルギー（音や内部エネルギーなど）に変化した場合は、2物体の質量や衝突前の速度に関わらず、衝突前に互いに近づく速さより衝突後に互いに遠ざかる速さの方が一般には小さい。このとき反発係数は値が1よりも小さくなる。

2物体が硬いほど値は1に近くなる。理想的な剛体では振動が生じ得ないので音も熱エネルギーも生じず衝突の前後で運動エネルギーの和が変化しないので、反発係数の値はe = 1となる。

衝突時に何らかの形で運動エネルギーが供給されない限り、反発係数が1よりも大きくなることはない。

定義式

物体1と物体2が衝突し速度がそれぞれv₁ からv₁' 、v₂ からv₂' に変わったとすると、反発係数e は

$e={\frac {|v'_{1}-v'_{2}|}{|v_{1}-v_{2}|}}=-{\frac {v'_{1}-v'_{2}}{v_{1}-v_{2}}}$

で定義される。

分類

e=1の衝突を弾性衝突 (次項参照)、0≦e<1の衝突を非弾性衝突といい、特にe=0の衝突を完全非弾性衝突という。

弾性衝突

反発係数が1となる衝突を（完全）弾性衝突という。弾性衝突では、運動量だけでなく運動エネルギーも保存している。

衝突で運動量と運動エネルギーの両方が保存するとき、反発係数が1になることは、以下のように示せる。

運動量保存の式として

$m_{1}v_{1}+m_{2}v_{2}=m_{1}v'_{1}+m_{2}v'_{2}$

運動エネルギー保存の式として

${\frac {1}{2}}m_{1}{v_{1}}^{2}+{\frac {1}{2}}m_{2}{v_{2}}^{2}={\frac {1}{2}}m_{1}{v'_{1}}^{2}+{\frac {1}{2}}m_{2}{v'_{2}}^{2}$

以上の２式を以下のように変形する。

$m_{1}(v_{1}-v'_{1})=m_{2}(v'_{2}-v_{2})$

$m_{1}({v_{1}}^{2}-{v'_{1}}^{2})=m_{2}({v'_{2}}^{2}-{v_{2}}^{2})$

衝突によってそれぞれの物体の速度が必ず変わると仮定すると、この２式より

$v_{1}+v'_{1}=v'_{2}+v_{2}$

よって

$v_{1}-v_{2}=-(v'_{1}-v'_{2})$

となり、これは衝突の前後で相対速度が同じ大きさで逆向きであることを示している。

したがって

$e=-{\frac {v'_{1}-v'_{2}}{v_{1}-v_{2}}}=1$

となる。このことが質量によらず成り立つことが特徴である。

例

• ビリヤードの球が硬いのは、反発係数を1に近づけるためである。

• スポーツの球技では、球の反発係数が重要となるため、ルールでかならず指定している。ただし、反発係数を直接指定するのは分かりにくいため、通常はその球技で用いる床面に、ある高さから球を落下させ、床面との衝突後、球がどの高さまで上がるかという形で間接的に指定している。はじめに高さh₁ からボールを落とし、衝突後に高さh₂ まではね上がったとすると、反発係数e は次式で求められる：

$e={\sqrt {\frac {h_{2}}{h_{1}}}}$

関連項目

• ボールの跳ね返り運動

{{デフォルトソート:はんはつけいすう}} [[Category:力学]] [[Category:無次元数]] [[de:Stoß (Physik)#Realer Stoß]]

A bouncing ball captured with a stroboscopic flash at 25 images per second: Ignoring air resistance, the square root of the ratio of the height of one bounce to that of the preceding bounce gives the coefficient of restitution for the ball/surface impact.

The coefficient of restitution (COR), also denoted by (e), is the ratio of the final to initial relative velocity between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectly inelastic collision has a coefficient of 0, but a 0 value does not have to be perfectly inelastic. It is measured in the Leeb rebound hardness test, expressed as 1000 times the COR, but it is only a valid COR for the test, not as a universal COR for the material being tested.

The value is almost always less than one due to initial translational kinetic energy being lost to rotational kinetic energy, plastic deformation, and heat. It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity.

${\text{Coefficient of restitution }}(e)={\frac {\left|{\text{Relative velocity after collision}}\right|}{\left|{\text{Relative velocity before collision}}\right|}}$

The mathematics were developed by Sir Isaac Newton in 1687. It is also known as Newton's experimental law.

Further details

Line of impact – It is the line along which e is defined or in absence of tangential reaction force between colliding surfaces, force of impact is shared along this line between bodies. During physical contact between bodies during impact its line along common normal to pair of surfaces in contact of colliding bodies. Hence e is defined as a dimensionless one-dimensional parameter.

Range of values for e – treated as a constant

e is usually a positive, real number between 0 and 1:

e = 0: This is a perfectly inelastic collision. This means kinetic energy along the common normal is 0. Kinetic energy is converted to heat or work done in deforming the objects.

0 < e < 1: This is a real-world inelastic collision, in which some kinetic energy is dissipated.

e = 1: This is a perfectly elastic collision, in which no kinetic energy is dissipated, and the objects rebound from one another with the same relative speed with which they approached.

e < 0: A COR less than zero would represent a collision in which the separation velocity of the objects has the same direction (sign) as the closing velocity, implying the objects passed through one another without fully engaging. This may also be thought of as an incomplete transfer of momentum. An example of this might be a small, dense object passing through a large, less dense one – e.g., a bullet passing through a target.

e > 1: This would represent a collision in which energy is released, for example, nitrocellulose billiard balls can literally explode at the point of impact. Also, some recent articles have described superelastic collisions in which it is argued that the COR can take a value greater than one in a special case of oblique collisions. These phenomena are due to the change of rebound trajectory caused by friction. In such collision kinetic energy is increased in a manner energy is released in some sort of explosion. It is possible that $e=\infty$ for a perfect explosion of a rigid system.

Maximum deformation phase – In any collision for 0 < e ≤ 1, there is a condition when for short moment along line of impact colliding bodies have same velocity when its condition of kinetic energy is lost in maximum fraction as heat, sound and light with deformation potential energy. For this short duration this collision e=0 and may be referred as inelastic phase.

Paired objects

The COR is a property of a pair of objects in a collision, not a single object. If a given object collides with two different objects, each collision would have its own COR. When an object is described as having a coefficient of restitution, as if it were an intrinsic property without reference to a second object, it is assumed to be between identical spheres or against a perfectly rigid wall.

A perfectly rigid wall is not possible but can be approximated by a steel block if investigating the COR of spheres with a much smaller modulus of elasticity. Otherwise, the COR will rise and then fall based on collision velocity in a more complicated manner.

Relationship with conservation of energy and momentum

In a one-dimensional collision, the two key principles are: conservation of energy (conservation of kinetic energy if the collision is perfectly elastic) and conservation of (linear) momentum. A third equation can be derived from these two, which is the restitution equation as stated above. When solving problems, any two of the three equations can be used. The advantage of using the restitution equation is that it sometimes provides a more convenient way to approach the problem.

Let $m_{1}$, $m_{2}$ be the mass of object 1 and object 2 respectively. Let $u_{1}$, $u_{2}$ be the initial velocity of object 1 and object 2 respectively. Let $v_{1}$, $v_{2}$ be the final velocity of object 1 and object 2 respectively.

${\begin{cases}{\frac {1}{2}}m_{1}u_{1}^{2}+{\frac {1}{2}}m_{2}u_{2}^{2}={\frac {1}{2}}m_{1}v_{1}^{2}+{\frac {1}{2}}m_{2}v_{2}^{2}\\m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}\end{cases}}$

From the first equation,

$m_{1}(u_{1}^{2}-v_{1}^{2})=m_{2}(v_{2}^{2}-u_{2}^{2})$

$m_{1}(u_{1}+v_{1})(u_{1}-v_{1})=m_{2}(v_{2}+u_{2})(v_{2}-u_{2})$

From the second equation,

$m_{1}(u_{1}-v_{1})=m_{2}(v_{2}-u_{2})$

After division,

$u_{1}+v_{1}=v_{2}+u_{2}$

$u_{1}-u_{2}=-(v_{1}-v_{2})$

${\frac {\left|v_{1}-v_{2}\right|}{\left|u_{1}-u_{2}\right|}}=1$

The equation above is the restitution equation, and the coefficient of restitution is 1, which is a perfectly elastic collision.

Sports equipment

「Bouncing ball#Sport regulations」も参照

The coefficient of restitution entered the common vocabulary, among golfers at least, when golf club manufacturers began making thin-faced drivers with a so-called "trampoline effect" that creates drives of a greater distance as a result of the flexing and subsequent release of stored energy, imparting greater impulse to the ball. The USGA (America's governing golfing body) has started testing drivers for COR and has placed the upper limit at 0.83. In April 2006 issued a more detailed report using five of the top golf balls used by professional golfers. In this report facts about golf balls beyond the subject of COR are highlighted. Due to the nature of polymers (man-made plastics) where rates of stress & strain are not Newtonian like fluids, metals etc. Because of this COR is a function of rates of clubhead speeds and diminish as clubhead speed increase. The USGA clearly states that nothing much can be gained beyond 90 mph clubhead speed. In the report COR ranges from 0.845 for 90 mph to as low as 0.797 at 130 mph. The above-mentioned "trampoline effect" clearly shows this since it reduces the rate of stress of the collision, or in another word "increases" the time of the collision. The number of this report; RB/cor2006-01by Steven J.Quintavalla Ph.D. According to one article (addressing COR in tennis racquets), "[f]or the Benchmark Conditions, the coefficient of restitution used is 0.85 for all racquets, eliminating the variables of string tension and frame stiffness which could add or subtract from the coefficient of restitution."

The International Table Tennis Federation specifies that the ball shall bounce up 24–26 cm when dropped from a height of 30.5 cm on to a standard steel block thereby having a COR of 0.887 to 0.923. For a hard linoleum floor with concrete underneath, a leather basketball has a COR around 0.81–0.85.

Equations

In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by:

$C_{R}={\frac {\left|v_{\text{b}}-v_{\text{a}}\right|}{\left|u_{\text{a}}-u_{\text{b}}\right|}}$, where:

$v_{\text{a}}$ is the final speed of object A after impact

$v_{\text{b}}$ is the final speed of object B after impact

$u_{\text{a}}$ is the initial speed of object A before impact

$u_{\text{b}}$ is the initial speed of object B before impact

Though $C_{R}$ does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. For two- and three-dimensional collisions of rigid bodies, the velocities used are the components perpendicular to the tangent line/plane at the point of contact, i.e. along the line of impact.

For an object bouncing off a stationary target, $C_{R}$ is defined as the ratio of the object's speed after the impact to that prior to impact:

$C_{R}={\frac {v}{u}}$, where

$v$ is the speed of the object after impact

$u$ is the speed of the object before impact

In a case where frictional forces can be neglected and the object is dropped from rest onto a horizontal surface, this is equivalent to:

$C_{R}={\sqrt {\frac {h}{H}}}$, where

$h$ is the bounce height

$H$ is the drop height

The coefficient of restitution can be thought of as a measure of the extent to which mechanical energy is conserved when an object bounces off a surface. In the case of an object bouncing off a stationary target, the change in gravitational potential energy, PE, during the course of the impact is essentially zero; thus, $C_{R}$ is a comparison between the kinetic energy, KE, of the object immediately before impact with that immediately after impact:

$C_{R}={\sqrt {\frac {KE_{\text{(after impact)}}}{KE_{\text{(before impact)}}}}}={\sqrt {\frac {{\frac {1}{2}}mv^{2}}{{\frac {1}{2}}mu^{2}}}}={\sqrt {\frac {v^{2}}{u^{2}}}}={\frac {v}{u}}$

In a cases where frictional forces can be neglected (nearly every student laboratory on this subject), and the object is dropped from rest onto a horizontal surface, the above is equivalent to a comparison between the PE of the object at the drop height with that at the bounce height. In this case, the change in KE is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus:

$C_{R}={\sqrt {\frac {PE_{\text{(at bounce height)}}}{PE_{\text{(at drop height)}}}}}={\sqrt {\frac {mgh}{mgH}}}={\sqrt {\frac {h}{H}}}$

Speeds after impact

The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between.

$v_{a}={\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{b}}C_{R}(u_{\text{b}}-u_{\text{a}})}{m_{\text{a}}+m_{\text{b}}}}$

and

$v_{\text{b}}={\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{a}}C_{R}(u_{\text{a}}-u_{\text{b}})}{m_{\text{a}}+m_{\text{b}}}}$

where

$v_{\text{a}}$ is the final velocity of the first object after impact

$v_{\text{b}}$ is the final velocity of the second object after impact

$u_{\text{a}}$ is the initial velocity of the first object before impact

$u_{\text{b}}$ is the initial velocity of the second object before impact

$m_{\text{a}}$ is the mass of the first object

$m_{\text{b}}$ is the mass of the second object

Derivation

The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). Using the notation from above where $u$ represents the velocity before the collision and $v$ after, yields:

${\begin{aligned}&m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}=m_{\text{a}}v_{\text{a}}+m_{\text{b}}v_{\text{b}}\\&C_{R}={\frac {\left|v_{\text{b}}-v_{\text{a}}\right|}{\left|u_{\text{a}}-u_{\text{b}}\right|}}\\\end{aligned}}$

Solving the momentum conservation equation for $v_{\text{a}}$ and the definition of the coefficient of restitution for $v_{\text{b}}$ yields:

${\begin{aligned}&{\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}-m_{\text{b}}v_{\text{b}}}{m_{\text{a}}}}=v_{\text{a}}\\&v_{\text{b}}=C_{R}(u_{\text{a}}-u_{\text{b}})+v_{\text{a}}\\\end{aligned}}$

Next, substitution into the first equation for $v_{\text{b}}$ and then resolving for $v_{\text{a}}$ gives:

${\begin{aligned}&{\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}-m_{\text{b}}C_{R}(u_{\text{a}}-u_{\text{b}})-m_{\text{b}}v_{\text{a}}}{m_{\text{a}}}}=v_{\text{a}}\\&\\&{\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{b}}C_{R}(u_{\text{b}}-u_{\text{a}})}{m_{\text{a}}}}=v_{\text{a}}\left[1+{\frac {m_{\text{b}}}{m_{\text{a}}}}\right]\\&\\&{\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{b}}C_{R}(u_{\text{b}}-u_{\text{a}})}{m_{\text{a}}+m_{\text{b}}}}=v_{\text{a}}\\\end{aligned}}$

A similar derivation yields the formula for $v_{\text{b}}$.

COR variation due to object shape and off-center collisions

When colliding objects do not have a direction of motion that is in-line with their centers of gravity and point of impact, or if their contact surfaces at that point are not perpendicular to that line, some energy that would have been available for the post-collision velocity difference will be lost to rotation and friction. Energy losses to vibration and the resulting sound are usually negligible.

Colliding different materials and practical measurement

When a soft object strikes a harder object, most of the energy available for the post-collision velocity will be stored in the soft object. The COR will depend on how efficient the soft object is at storing the energy in compression without losing it to heat and plastic deformation. A rubber ball will bounce better off concrete than a glass ball, but the COR of glass-on-glass is a lot higher than rubber-on-rubber because some of the energy in rubber is lost to heat when it is compressed. When a rubber ball collides with a glass ball, the COR will depend entirely on the rubber. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material.

Since there is no perfectly rigid material, hard materials such as metals and ceramics have their COR theoretically determined by considering the collision between identical spheres. In practice, a 2-ball Newton's cradle may be employed but such a set up is not conducive to quickly testing samples.

The Leeb rebound hardness test is the only commonly-available test related to determining the COR. It uses a tip of tungsten carbide, one of the hardest substances available, dropped onto test samples from a specific height. But the shape of the tip, the velocity of impact, and the tungsten carbide are all variables that affect the result that is expressed in terms of 1000*COR. It does not give an objective COR for the material that is independent from the test.

A comprehensive study of coefficients of restitution in dependence on material properties (elastic moduli, rheology), direction of impact, coefficient of friction and adhesive properties of impacting bodies can be found in.

Predicting from material properties

The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. To avoid the complications of rotational and frictional losses, we can consider the ideal case of an identical pair of spherical objects, colliding so that their centers of mass and relative velocity are all in-line.

Many materials like metals and ceramics (but not rubbers and plastics) are assumed to be perfectly elastic when their yield strength is not approached during impact. The impact energy is theoretically stored only in the spring-effect of elastic compression and results in e = 1. But this applies only at velocities less than about 0.1 m/s to 1 m/s. The elastic range can be exceeded at higher velocities because all the kinetic energy is concentrated at the point of impact. Specifically, the yield strength is usually exceeded in part of the contact area, losing energy to plastic deformation by not remaining in the elastic region. To account for this, the following estimates the COR by estimating the percent of the initial impact energy that did not get lost to plastic deformation. Approximately, it divides how easy a volume of the material can store energy in compression ($1/{\text{elastic modulus}}$) by how well it can stay in the elastic range ($1/{\text{yield strength}}$):

$\%{\text{impact energy available for restitution}}\propto {\frac {\text{yield strength}}{\text{elastic modulus}}}$

For a given material density and velocity this results in:

${\text{coefficient of restitution}}\propto {\sqrt {\frac {\text{yield strength}}{\text{elastic modulus}}}}$

A high yield strength allows more of the "contact volume" of the material to stay in the elastic region at higher energies. A lower elastic modulus allows a larger contact area to develop during impact so the energy is distributed to a larger volume beneath the surface at the contact point. This helps prevent the yield strength from being exceeded.

A more precise theoretical development shows the velocity and density of the material to also be important when predicting the COR at moderate velocities faster than elastic collision (greater than 0.1 m/s for metals) and slower than large permanent plastic deformation (less than 100 m/s). A lower velocity increases the coefficient by needing less energy to be absorbed. A lower density also means less initial energy needs to be absorbed. The density instead of mass is used because the volume of the sphere cancels out with the volume of the affected volume at the contact area. In this way, the radius of the sphere does not affect the coefficient. A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by $\left({\frac {R_{1}}{R_{2}}}\right)^{\frac {3}{8}}$

Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material.

• e = coefficient of restitution
• S_y = dynamic yield strength (dynamic "elastic limit")
• E′ = effective elastic modulus
• ρ = density
• v = velocity at impact
• μ = Poisson's ratio

$e=3.1\left({\frac {S_{\text{y}}}{1}}\right)^{\frac {5}{8}}\left({\frac {1}{E'}}\right)^{\frac {1}{2}}\left({\frac {1}{v}}\right)^{\frac {1}{4}}\left({\frac {1}{\rho }}\right)^{\frac {1}{8}}$

$E'={\frac {E}{1-\mu ^{2}}}$

This equation overestimates the actual COR. For metals, it applies when v is approximately between 0.1 m/s and 100 m/s and in general when:

$0.001<{\frac {\rho v^{2}}{S_{\text{y}}}}<0.1$

At slower velocities the COR is higher than the above equation predicts, theoretically reaching e=1 when the above fraction is less than $10^{-6}$ m/s. It gives the following theoretical coefficient of restitution for solid spheres dropped 1 meter (v = 4.5 m/s). Values greater than 1 indicate that the equation has errors. Yield strength instead of dynamic yield strength was used.

Metals and Ceramics:	Predicted COR, e
silicon	1.79
Alumina	0.45 to 1.63
silicon nitride	0.38 to 1.63
silicon carbide	0.47 to 1.31
highest amorphous metal	1.27
tungsten carbide	0.73 to 1.13
stainless steel	0.63 to 0.93
magnesium alloys	0.5 to 0.89
titanium alloy grade 5	0.84
aluminum alloy 7075-T6	0.75
glass (soda-lime)	0.69
glass (borosilicate)	0.66
nickel alloys	0.15 to 0.70
zinc alloys	0.21 to 0.62
cast iron	0.3 to 0.6
copper alloys	0.15 to 0.55
titanium grade 2	0.46
tungsten	0.37
aluminum alloys 3003 6061, 7075-0	0.35
zinc	0.21
nickel	0.15
copper	0.15
aluminum	0.1
lead	0.08

The COR for plastics and rubbers are greater than their actual values because they do not behave as ideally elastic as metals, glasses, and ceramics due to heating during compression. So the following is only a guide to ranking of polymers.

Polymers (overestimated compared to metals and ceramics):

• polybutadiene (golf balls shell)
• butyl rubber
• EVA
• silicone elastomers
• polycarbonate
• nylon
• polyethylene
• Teflon
• polypropylene
• ABS
• acrylic
• PET
• polystyrene
• PVC

For metals the range of speeds to which this theory can apply is about 0.1 to 5 m/s which is a drop of 0.5 mm to 1.25 meters (page 366).

See also

• Bouncing ball
• Collision
• Resilience

References

Works cited

• Cross, Rod (2006). The bounce of a ball. Physics Department, University of Sydney, Australia. http://www.physics.usyd.edu.au/~cross/PUBLICATIONS/BallBounce.pdf 2008年1月16日閲覧。. 
• Walker, Jearl (2011). Fundamentals Of Physics (9th ed.). David Halliday, Robert Resnick, Jearl Walker. ISBN 978-0-470-56473-8 

External links

• Wolfram Article on COR
• Bennett & Meepagala (2006年). “Coefficients of Restitution”. The Physics Factbook.
• Chris Hecker's physics introduction
• "Getting an extra bounce" by Chelsea Wald
• FIFA Quality Concepts for Footballs – Uniform Rebound
• Bowley, Roger (2009年). “Coefficient of Restitution”. Sixty Symbols. Brady Haran for the University of Nottingham.

{{DEFAULTSORT:Coefficient Of Restitution}} [[Category:Mechanics]] [[Category:Classical mechanics]] [[Category:Ratios]] [[de:Stoß (Physik)#Realer Stoß]]