Chapter 9 Linear Momentum and Collisions

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Chapter 9Linear Momentum and Collisions

• Linear momentum is defined as
• p mv
• Momentum is given by mass times velocity.
• Momentum is a vector.
• The units of momentum are (no special unit)
• p kgm/s

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• Since p is a vector, we can also consider the
  components of momentum
• px mvx
• py mvy
• pz mvz
• Note momentum is large if m and/or v is large.
  (define large, meaning hard for you to stop).
• Name an object with large momentum but small
  velocity.
• Name an object with large momentum but small mass

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• Recall that

Another way of writing Newtons Second Law is F
Dp/Dt rate of change of momentum This form is
valid even if the mass is changing. This form is
valid even in Relativity and Quantum Mechanics.
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Impulse

• We can rewrite F Dp/Dt as
• FDt Dp
• I FDt is known as the impulse.
• The impulse of the force acting on an object
  equals the change in the momentum of that object.
• Exercise Show that impulse and momentum have
  the same units.

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Example

• A 0.3-kg hockey puck moves on frictionless ice at
  8 m/s toward the wall. It bounces back away from
  the wall at 5 m/s. The puck is in contact with
  the wall for 0.2 s.
• What is the change in momentum of the hockey puck
  during the bounce?
• What is the impulse on the hockey puck during the
  bounce?
• What is the average force of the wall on the
  hockey puck during the bounce?

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• If there are no external forces on a system, then
  the total momentum of that system is constant.
  This is known as
• The Principle of
• Conservation
• of
• Momentum
• In that case, pi pf.

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Conservation of Momentum and Newtons Third Law

• Consider a system consisting of just the two
  masses m1 and m2.
• Mass m1 exerts a force F21 on mass m2.
• Mass m2 exerts a force F12 on mass m1.
• Force on m1 rate of change of momentum of m1
• F12 Dp1 / Dt
• Force on m2 rate of change of momentum of m2
• F21 Dp2 / Dt
• Dp1 / Dt Dp2 / Dt F12 F21 0 (Newtons
  Third Law).
• D(p1p2 )/ Dt 0
• Rate of change of total momentum is zero.
• Total Momentum does not change if net external
  force is zero

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Internal vs. External Forces
Here the system is just the box and table. Any
forces between those two objects are internal.
Example The normal forces between the table and
the box are internal forces. Internal forces on
the system sum to zero.
system
External forces do not necessarily sum to zero.
Something outside the circle is pushing or
pulling something inside the circle. Example
gravity is an external force.
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Example

• Two skaters are standing on frictionless ice.
  Skater A has a mass of 50 kg and skater B has a
  mass of 80 kg. Skater A pushes Skater B for 0.25
  s, causing Skater B to move away at 10 m/s.
• The force of gravity, and the normal force from
  the ice on each skater are EXTERNAL forces.
  However, for each skater, the external forces add
  to zero.
• What is the velocity of Skater A after he pushes
  Skater B?
• What is the change in momentum of Skater B?
• What is the average force exerted on Skater B by
  Skater A during the 0.2 s push?
• (d) What is the change in momentum of Skater A?
• (e) What is the average force exerted on Skater A
  by Skater B during the 0.2 s?

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Walker, Problem 16, pg. 266
A 0.175-g bee walks on a stick floating in the
water. If the bee walks with a speed of 1.41
cm/s relative to the still water, what is the
speed of the 4.75-g stick relative to the still
water? Neglect the friction of the stick in the
water. Do not neglect the friction of the bee
walking on the stick!
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Collisions

• In general, a collision is an interaction in
  which
• two objects strike one another
• the net external impulse is zero or negligibly
  small (momentum is conserved)
• Examples car crash billiard balls
• Collisions can involve more than 2 objects

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• From the conservation of momentum
• pi pf
• m1v1,i m2v2,i m1v1,f m2v2,f

v2,i
v1,i
v2,f
v1,f
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• What about conservation of energy?
• We said earlier that the total energy of an
  isolated system is conserved, but the total
  kinetic energy may change.
• elastic collisions K is conserved
• inelastic collisions K is not conserved
• perfectly inelastic objects stick together after
  colliding

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Perfectly Inelastic Collisions

• After a perfectly inelastic collision the two
  objects stick together and move with the same
  final velocity
• pi pf
• m1 v1,i m2 v2,i (m1 m2)vf

This gives the maximum possible loss of kinetic
energy. In non-relativistic collisions, the total
mass is conserved
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Walker, Problem 26, pg. 267
A 0.470-kg block of wood hangs from the ceiling
by a string, and a 0.0700-kg wad of putty is
thrown straight upward, striking the bottom of
the block with a speed of 5.60 m/s. The wad of
putty sticks to the block. (a) Is the mechanical
energy of the putty-block system conserved in the
collision? (b) How high does the putty-block
system rise above the original position of the
block?
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Elastic Collisions

• Kinetic energy is conserved in addition to
  momentum
• pi pf
• Ki Kf

Lots of variables, keep track!
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Walker, Problem 33, pg. 267
A charging bull elephant with a mass of 5400 kg
comes directly toward you with a speed of 4.3
m/s. You toss a 0.150-kg rubber ball at the
elephant with a speed of 8.11 m/s. (a) When the
ball bounces back toward you, what is its speed?
(b) How do you account for the fact that the
balls kinetic energy has increased?
Drop a tennis ball together with a
basketball. The tennis ball will bounce back 9
times higher than your initial height (if it
doesnt hit you in the face first!).
g
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Elastic Collisions in 1-dimension

• Kinetic energy is conserved in addition to
  momentum
• pi pf
• Ki Kf

Relative velocity of approach before collision
relative velocity of separation after collision
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Center of Mass
The center of mass (CM) of an object or a group
of objects (system) is the average location of
the mass in the system. The system behaves as if
all of its mass were concentrated at the center
of mass.
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The center of mass is not always located on the
object.
Where is the CM for this object?
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Calculating the Center of Mass
For two objects
In general, X coordinate of center of mass
In general, Y coordinate of center of mass
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Motion of the Center of Mass
A system of objects behaves as if all its mass
were located at the center of mass.
Velocity of CM
Acceleration of CM
MAcm Fnet,ext
Columbia re-entry disaster.
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Walker Problem 34, pg. 267
Find the x coordinate of the center of mass of
the bricks shown in the Figure.
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Infinite Mayan Arch

• nth block is stacked a distance L/n from the edge
  of last block.
• No matter how many blocks are stacked, the Center
  of Mass remains at xltL.
• Stack is stable (barely)
• Series (1/2) (1/3) (1/4) increases without
  limit.
• With enough blocks, the nth block can be as far
  from x0 as you want!!

L/5
L/4
L/3
L/2
L
0
x
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Example

• Car A has a mass of 1000 kg and is traveling to
  the right at 5 m/s. Car B has a mass of 5000 kg
  and is traveling to the right at 3 m/s. Car A
  collides with Car B and they stick together.
• What is the velocity of the center of mass before
  the collision?
• What is the velocity of the center of mass after
  the collision?