Further details Coefficient of restitution

1 further details

1.1 range of values e – treated constant
1.2 paired objects
1.3 relationship conservation of energy , momentum
1.4 sports equipment

further details

line of impact – line along e defined or in absence of tangential reaction force between colliding surfaces, force of impact shared along line between bodies. during physical contact between bodies during impact line along common normal pair of surfaces in contact of colliding bodies. hence e defined dimensionless one-dimensional parameter.

range of values e – treated constant

e positive, real number between 0 , 1:

e = 0: inelastic collision. objects not move apart after collision, instead coalesce. kinetic energy converted heat or work done in deforming objects.

0 < e < 1: real-world inelastic collision, in kinetic energy dissipated.

e = 1: elastic collision, in no kinetic energy dissipated, , objects rebound 1 same relative speed approached.

e < 0: cor less 0 represent collision in separation velocity of objects has same direction (sign) closing velocity, implying objects passed through 1 without engaging. may thought of incomplete transfer of momentum. example of might small, dense object passing through large, less dense 1 – e.g., bullet passing through target, or motorcycle passing through motor home or wave tearing through dam.

e > 1: represent collision in energy released, example, nitrocellulose billiard balls can literally explode @ point of impact. also, recent articles have described superelastic collisions in argued cor can take value greater 1 in special case of oblique collisions. these phenomena due change of rebound trajectory caused friction. in such collision kinetic energy increased in manner energy released in sort of explosion. possible



e
=
∞


{\displaystyle e=\infty }

perfect explosion of rigid system.

maximum deformation phase – in collision 0 < e ≤ 1, there condition when short moment along line of impact colliding bodies have same velocity when condition of kinetic energy lost in maximum fraction heat, sound , light deformation potential energy. short duration collision e=0 , may referred inelastic phase.

paired objects

the cor property of pair of objects in collision, not single object. if given object collides 2 different objects, each collision have own cor. when object described having coefficient of restitution, if intrinsic property without reference second object, assumed between identical spheres or against rigid wall.

a rigid wall not possibly can approximated steel block if investigating cor of spheres smaller modulus of elasticity. otherwise, cor rise , fall based on collision velocity in more complicated manner.

relationship conservation of energy , momentum

in one-dimensional collision, 2 key principles are: conservation of energy (conservation of kinetic energy if collision elastic) , conservation of (linear) momentum. third equation can derived these two, restitution equation stated above. when solving problems, 2 of 3 equations can used. advantage of using restitution equation provides more convenient way approach problem.

let




m

1




{\displaystyle m_{1}}

,




m

2




{\displaystyle m_{2}}

mass of object 1 , object 2 respectively. let




u

1




{\displaystyle u_{1}}

,




u

2




{\displaystyle u_{2}}

initial velocity of object 1 , object 2 respectively. let




v

1




{\displaystyle v_{1}}

,




v

2




{\displaystyle v_{2}}

final velocity of object 1 , object 2 respectively.

{





1
2



m

1



u

1


2


+


1
2



m

2



u

2


2


=


1
2



m

1



v

1


2


+


1
2



m

2



v

2


2







m

1



u

1


+

m

2



u

2


=

m

1



v

1


+

m

2



v

2










{\displaystyle {\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 first equation,

m

1


(

u

1


2


−

v

1


2


)
=

m

2


(

v

2


2


−

u

2


2


)


{\displaystyle 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


)


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

from second equation,

m

1


(

u

1


−

v

1


)
=

m

2


(

v

2


−

u

2


)


{\displaystyle m_{1}(u_{1}-v_{1})=m_{2}(v_{2}-u_{2})}

after division,

u

1


+

v

1


=

v

2


+

u

2




{\displaystyle u_{1}+v_{1}=v_{2}+u_{2}}







u

1


−

u

2


=
−
(

v

1


−

v

2


)


{\displaystyle u_{1}-u_{2}=-(v_{1}-v_{2})}






−




v

1


−

v

2





u

1


−

u

2





=
1


{\displaystyle -{\frac {v_{1}-v_{2}}{u_{1}-u_{2}}}=1}

the equation above restitution equation, , coefficient of restitution 1, elastic collision.

sports equipment

the coefficient of restitution entered common vocabulary, among golfers @ least, when golf club manufacturers began making thin-faced drivers so-called trampoline effect creates drives of greater distance result of flexing , subsequent release of stored energy, imparting greater impulse ball. usga (america s governing golfing body) has started testing drivers cor , has placed upper limit @ 0.83. according 1 article (addressing cor in tennis racquets), [f]or benchmark conditions, coefficient of restitution used 0.85 racquets, eliminating variables of string tension , frame stiffness add or subtract coefficient of restitution.

the international table tennis federation specifies ball shall bounce 24–26 cm when dropped height of 30.5 cm on standard steel block thereby having cor of 0.89 0.92. hard linoleum floor concrete underneath, leather basketball has cor around 0.81–0.85.