,
Momentum & Impulse : Coefft. Restitution
 

[ coefft. restitution][ oblique collisions]

 

 

 

 

The Coefficient of Restitution

The Coefficient of Restitution (e) is a variable number with no units, with limits from zero to one.

0 ≤ e ≤ 1

'e' is a consequence of Newton's Experimental Law of Impact, which describes how the speed of separation of two impacting bodies compares with their speed of approach.

note: the speeds are relative speeds

coefficient of restitution

If we consider the speed of individual masses before and after collision, we obtain another useful equation:

uA = initial speed of mass A

uB = initial speed of mass B

vA = final speed of mass A

vB = final speed of mass B

relative initial speed of mass A to mass B = uB - uA

relative final speed of mass A to mass B = vB - vA

coefficient of restitution equation

note: in this equation the absolute of uB - uA and vB - vA are used ( |absolute| no net negative result )

 

Example

A 5 kg mass moving at 6 ms-1 makes a head-on collision with a 4 kg mass travelling at 3 ms-1 .
Assuming that there are no external forces acting on the system, what are the velocities of the two masses after impact?
(assume coefficient of restitution e = 0.5 )

uA = initial speed of 5 kg mass (mass A)= 6 ms-1

uB = initial speed of 4 kg mass (mass B)= 3 ms-1

mA = 5 kg       mB = 4 kg

vA = final speed of mass A        vB = final speed of mass B

 

momentum before the collision equals momentum after

hence,        mA uA + mB uB = mA vA + mB vB

also

coefficient of restitution equation

substituting for e, mA , uA , mB , uB

we obtain two simultaneous equations

from the conservation of momentum,

5 x 0.6 + 4(-3) = 5 vA + 4 vB

3 - 12 = 5 vA + 4 vB

-9 = 5 vA + 4 vB

5 vA + 4 vB = -9        (i

from the coefficient of restitution expression,

restitution problem#01

0.5(uA - uB) = vB - vA

(0.5 x 6) - (0.5(-3)) = vB - vA

3 + 1.5 = vB - vA

4.5 = vB - vA

vB - vA = 4.5        (ii

multiplying (ii by 5 and adding

5 vA + 4 vB = -9

- 5 vA + 5 vB = 22.5

9 vB = 13.5

vB = 1.5 ms-1

from (ii

1.5 - vA = 4.5

vA = 1.5 - 4.5 = -3

vA = -3 ms-1

Ans. The velocities of the 5 kg and 4 kg masses are -3 ms-1and 1.5 ms-1, respectively.

 

back to top

Oblique Collisions

For two masses colliding along a line, Newton's Experimental law is true for component speeds. That is, the law is applied twice: to each pair of component speeds acting in a particular direction.

Example

A particle of mass m impacts a smooth wall at 4u ms-1 at an angle of 30 deg. to the vertical. The particle rebounds with a speed ku at 90 deg. to the original direction and in the same plane as the impact trajectory.

What is:
i) the value of the constant 'k' ?
ii) the coefficient of restitution between the wall and the particle?
iii) the magnitude of the impulse of the wall on the particle

i) There is no momentum change parallel to the wall.

coefft. rest prob 01

 

ii) The coefficient of restitution'e' is the ratio of the speed of separation to the speed of approach:

problem part (ii

 

iii) The impulse is the change of momentum.

Since the vertical unit vectors are unchanged, the momentum change just concerns the horizontal vector components.

hence,

problem #1 part (iii

back to top

 

your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone

[ PURE MATHS ][ MECHANICS ][ STATISTICS ]

 

VIDEO

linear motion
acceleration
average velocity
instantaneous velocity
circular motion 1
projectile motion
parabolic motion
relative motion
Newton's 2nd Law
connected particles
KE & PE changes
power
elastic strings
conservation momtm.
coefft. restitution
 
 

INTERACTIVE

 
 

EXAM PAPERS(.pdf)

EdxlM1 Mechanics spec.
EdxlM1 Mechanics ans.
EdxlM2 Mechanics spec.
EdxlM2 Mechanics ans.
EdxlM3 Mechanics spec.
EdxlM3 Mechanics ans.
EdxlM4 Mechanics spec.
EdxlM4 Mechanics ans.
EdxlM5 Mechanics spec.
EdxlM5 Mechanics ans.
 

TOPIC NOTES(.pdf)

 
uniform acceleration
non-uniform accln.
simple harmonic motion
projectiles
circular motion
relative motion
Newton's Laws
connected particles
work & energy
power & efficiency
more circular motion
elastic strings
impulse
conservation momtm
coeff. restitution
part. forces in equilib.
friction
rigid bodies
MORE...
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Google