Download 8-3 Perfectly Inelastic Collisions

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Physics for Scientists and
Engineers, 3rd edition
Fishbane
Gasiorowicz
Thornton
© 2005 Pearson Prentice Hall
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Chapter 8
Linear Momentum, Collisions,
and the Center of Mass
Main Points of Chapter 8
• Definition of momentum
• Conservation of momentum
• Collisions – elastic and inelastic
• Impulse
• Explosions
• Center of mass
• Rocket motion
8-1 Momentum and its Conservation
Definition of linear momentum:
(8-2)
Writing force and kinetic energy in
terms of momentum:
(8-3)
and
(8-4)
8-1 Momentum and its Conservation
Conservation of momentum:
for zero net external force,
(8-9)
• All internal forces are action-reaction
pairs – momentum is conserved
8-2 Collisions and Impulse
Impulsive forces
• A force often acts on an object for a limited time
• The integral of the force over time gives the
change in momentum, and is called the impulse
8-2 Collisions and Impulse
Definition of
impulse:
(8-14)
Change in
momentum:
(8-15)
• The same change in momentum can be
created by a large force acting for a short time,
or a smaller force acting for a longer time
8-2 Collisions and Impulse
Classification of Collisions
1. Objects can stick together
after collision
2. Objects can remain
unchanged
3. Mass can be transferred
from one object to another
4. One or both objects can
shatter
8-3 Perfectly Inelastic Collisions; Explosions
Perfectly Inelastic Collisions:
• Objects stick together after colliding
• Only one final velocity
(8-20)
8-3 Perfectly Inelastic Collisions;
Explosions
Energy Loss in Perfectly Inelastic
Collisions
• Momentum is conserved
• Kinetic energy is not
(8-24)
8-3 Perfectly Inelastic Collisions; Explosions
Explosions
• Perfectly inelastic collision “run in
reverse”
• Most easily analyzed in reference frame
where initial object is at rest
• In that case, total momentum is always
zero:
(8-25)
8-4 Elastic Two-Body Collisions in One
Dimension
• Elastic collisions conserve both momentum
and kinetic energy
• The relative velocity of the colliding objects
changes sign but does not change magnitude.
8-4 Elastic Two-Body Collisions in One
Dimension
When Object 2 is
initially at rest:
(8-35b)
(8-35a)
When the initial total
momentum is zero:
(8-38)
8-5 Elastic Collisions in Two and Three
Dimensions
In the general case, given masses and
initial velocities, there are more variables
than equations – no unique solution
8-5 Elastic Collisions in Two and Three
Dimensions
• Specific cases can be solved, given
enough information
Conservation of Momentum:
Conservation of Energy:
(8-41)
(8-42)
8-6 Center of Mass
• Importance: objects move under
Newton’s second law as though all
mass were at center of mass
• How to find it: Center of mass is
“average” position of mass in all three
dimensions
• Balance point
• Independent of coordinate system
8-6 Center of Mass
Center of mass of three pointlike
objects:
8-6 Center of Mass
Center of Mass Motion in the
Absence of External Forces
(8-50)
In the absence of external
forces the center of mass
moves with constant velocity.
8-6 Center of Mass
Center of Mass Motion in the
Presence of External Forces
(8-55)
The object moves under Newton’s second law
as though all its mass were at center of mass.
8-6 Center of Mass
Center of Mass of a Continuous
Mass Distribution
For a continuous distribution,
integrate over the whole mass:
(8-63)
8-6 Center of Mass
How to deal with holes:
1. Find the center of mass of
the piece with no hole.
2. Find the center of mass of
the hole.
3. Subtract.
8-7 Rocket Motion
• Similar to an explosion except that
mass is continually lost
• Derivative accounts for lost mass
(8-69)
Summary of Chapter 8
• Momentum:
(8-2)
• Momentum is conserved in the absence of
external forces
• Impulse is force integrated over time; is
also change in momentum
• Perfectly inelastic collision: objects stick
together afterwards. Conserves momentum
but not energy.
• Explosion: perfectly inelastic collision run
backwards
Summary of Chapter 8, cont.
• Elastic collisions: conserve
momentum and energy
• Center of mass: object obeys
Newton’s 2nd law as though all mass
were there
• Sum of mass x distance from a
given point
• Center of mass of a continuous
object: find by integrating
Summary of Chapter 8, cont.
• Center of mass of object with hole: treat
hole as having “negative mass”
• Rocket motion: mass is expelled
continuously; need to find speed at any
given time by differentiating