6. Collision for PHY103.ppt

Conservation of Momentum
 In an isolated and closed system,
the total momentum of the
system remains constant in time.
 Isolated system: no external forces
 Closed system: no mass enters or
leaves
 The linear momentum of each
colliding body may change
 The total momentum P of the system
cannot change.
April
12,
2023

 Start from impulse-momentum
theorem
 Since
 Then
 So
i
f v
m
v
m
t
F 2
2
2
2
12






i
f v
m
v
m
t
F 1
1
1
1
21






April
12,
2023
t
F
t
F 


 12
21


)
( 2
2
2
2
1
1
1
1 i
f
i
f v
m
v
m
v
m
v
m








f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1








 When no external forces act on a system consisting of
two objects that collide with each other, the total
momentum of the system remains constant in time
 When then
 For an isolated system
 Specifically, the total momentum before the collision
will equal the total momentum after the collision
f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1







April
12,
2023
i
f
net p
p
p
t
F









0

p

0

net
F

i
f p
p




Types of Collisions
 Momentum is conserved in any collision
 Inelastic collisions: rubber ball and hard ball
 Kinetic energy is not conserved
 Perfectly inelastic collisions occur when the objects
stick together
 Elastic collisions: billiard ball
 both momentum and kinetic energy are conserved
 Actual collisions
 Most collisions fall between elastic and perfectly
inelastic collisions
April
12,
2023

More About Elastic Collisions
 Both momentum and kinetic energy
are conserved
 Typically have two unknowns
 Momentum is a vector quantity
 Direction is important
 Be sure to have the correct signs
 Solve the equations simultaneously
April
12,
2023
2
2
2
2
1
1
2
2
2
2
1
1
2
2
1
1
2
2
1
1
2
1
2
1
2
1
2
1
f
f
i
i
f
f
i
i
v
m
v
m
v
m
v
m
v
m
v
m
v
m
v
m







Elastic Collisions
 A simpler equation can be used in place of the KE
equation
i
f
f
i v
v
v
v 2
2
1
1 


f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1 


April
12,
2023
)
v
v
(
v
v f
2
f
1
i
2
i
1 



2
2
2
2
1
1
2
2
2
2
1
1
2
1
2
1
2
1
2
1
f
f
i
i v
m
v
m
v
m
v
m 


)
)(
(
)
)(
( 2
2
2
2
2
1
1
1
1
1 i
f
i
f
f
i
f
i v
v
v
v
m
v
v
v
v
m 




)
(
)
( 2
2
2
1
1
1 i
f
f
i v
v
m
v
v
m 


)
(
)
( 2
2
2
2
2
2
1
2
1
1 i
f
f
i v
v
m
v
v
m 


f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1 



More about Perfectly Inelastic
Collisions
 When two objects stick together
after the collision, they have
undergone a perfectly inelastic
collision
 Conservation of momentum
 Kinetic energy is NOT conserved
2
1
2
2
1
1
m
m
v
m
v
m
v i
i
f



April
12,
2023
f
i
i v
m
m
v
m
v
m )
( 2
1
2
2
1
1 



Collisions Summary
 In an elastic collision, both momentum and kinetic energy
are conserved
 In a non-perfect inelastic collision, momentum is conserved
but kinetic energy is not. Moreover, the objects do not stick
together
 In a perfectly inelastic collision, momentum is conserved,
kinetic energy is not, and the two objects stick together
after the collision, so their final velocities are the same
 Elastic and perfectly inelastic collisions are limiting cases,
most actual collisions fall in between these two types
 Momentum is conserved in all collisions
April
12,
2023

Summary of Types of Collisions
 In an elastic collision, both momentum and kinetic
energy are conserved
 In an inelastic collision, momentum is conserved but
kinetic energy is not
 In a perfectly inelastic collision, momentum is
conserved, kinetic energy is not, and the two objects
stick together after the collision, so their final
velocities are the same
April
12,
2023
i
f
f
i v
v
v
v 2
2
1
1 

 f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1 


f
f
i
i v
m
v
m
v
m
v
m 2
2
1
1
2
2
1
1 


f
i
i v
m
m
v
m
v
m )
( 2
1
2
2
1
1 



6. Collision for PHY103.ppt

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