CONSERVATION OF MOMENTUM
1. Momentum is the product of mass and velocity. It is a vector quantity and its direction is towards
the direction of the velocity.
2. Impulse is the product of force and time during which the force acts. The impulse is equal to the
change in momentum.
3. Elastic bodies are bodies which return to their original shapes after a temporary deformation during
collision.
4. Law of Conservation of momentum. If tow or more bodies interact, the momentum after the
interactions is equal to the momentum before interaction. The total momentum of any system of
bodies is unchanged by any interactions between the different members of the system.
5. Resilience is the ability of a body to undergo compression or rapid deformation without the
development of permanent deformation.
6. Restitution is the vigor (energy) with which a body restores to its original shape and size after
deformation.
7. Coefficient of restitution ( r ) the ratio of the velocity with which the two bodies separate after
collision to the velocity of approach before collision.
r = 1 for perfectly elastic collision
r = 0 for perfectly inelastic collision
r = between 0 – 1 for semi-elastic collision.
COLLISION PHENOMENA
1. In a perfectly elastic collision, kinetic energy as well as momentum is conserved. The velocity of approach is equal to the velocity of separation in magnitude but opposite in direction. ( NOTE : No perfectly elastic collision for macroscopic bodies ). Ex. Collision between atomic nuclei, atoms, molecules and electrons
2. Inelastic collision is any collision for which the final kinetic energy is less than the initial kinetic energy.
3. Perfectly inelastic collision is any collision wherein the two colliding bodies sticks together indefinitely. The colliding bodies are permanently deformed and never separate, hence both have the same final velocity and the velocity of separation is zero since the two bodies sticks together.
4. Semi-elastic collision is a type of collision that is not perfectly elastic.
NOTE :
In every collision momentum is conserved, kinetic energy is conserved only in perfectly elastic collision.
MOMENTUM EQUATIONS
1. p = mv , p => momentum in N-s , m => mass in kg ,
v => velocity in m/s
2. I = Ft , F => force in newton , t => time in second
Head-on Collision
During collision
F1 = F2
m1a1 = m2a2
where a1 = u1 – v1 . and a2 = v2 – u2 .
t t
m1 ( u1 – v1 ) = m2 ( v2 – u2 )
t t
m1u1 – m1v1 = m2v2 – m2u2
which simplifies to
3. m1u1 + m2u2 = m2u2 + m2v2
( this is now the statement of the law of momentum)
momentum before collision = momentum after collision
4 . v2 = m1u1 + m2u2 – m1v1
m2
5. Kinetic energy before collision Ekb = ½ ( m1u12 + m2u22 )
6. Kinetic energy after collision Eka = ½ (m1v12 + m2v22 )
7. Energy loss during collision EL = Ekb – Eka
8. Net velocity of approach ∑ u = u1 – u2
9. Relative velocity of separation ∑ v = – ( v1 – v2 ) or v2 – v1
10. Coefficient of restitution
r = – v1 – v2 ==> r ( u1 – u2 ) = v2 – v1
u1 – u2
11. v2 = r ( u1 – u2 ) + v1
Substitute equation 11 in equation 3
m1u1 + m2u2 = m2u2 + m2 [r ( u1 – u2 ) + v1 ]
m1u1 + m2u2 = m2u2 + m2 r u1 – m2 r u2 + m2 v1
m1u1 + m2u2 – m2 r u1 + m2 r u2 = v1 ( m1 + m2 )
add m2 u1 – m2 u1 on the left side of the equation
m1u1 + m2u2 + m2 u1 – m2 u1 – m2 r u1 + m2 r u2 = v1 ( m1 + m2 )
u1 ( m1 + m2 ) – m2 ( u1 – u2 + r u1 – r u2 ) = v1 ( m1 + m2 )
u1 ( m1 + m2 ) – m2 ( u1 – u2 ) ( 1 + r ) = v1 ( m1 + m2 )
12. v1 = u1 ( m1 + m2 ) – m2 ( u1 – u2 )( 1 + r )
m1 + m2
and similarly
13. v2 = u2 ( m1 + m2 ) – m1 ( u1 – u2 )( 1 + r )
m1 + m2
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