Linear Momentum Lecturer: Professor Stephen T. Thornton

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Linear Momentum Lecturer Professor Stephen
T. Thornton
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Reading Quiz
A system of particles is known to have a total
kinetic energy of zero. What can you say about
the total momentum of the system?

• A) momentum of the system is positive
• B) momentum of the system is negative
• C) momentum of the system is zero
• you cannot say anything about the
• momentum of the system

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Reading Quiz
A system of particles is known to have a total
kinetic energy of zero. What can you say about
the total momentum of the system?

• A) momentum of the system is positive
• B) momentum of the system is negative
• C) momentum of the system is zero
• D) you cannot say anything about the momentum of
  the system

Since the total kinetic energy is zero, this
means that all of the particles are at rest (v
0). Therefore, since nothing is moving, the
total momentum of the system must also be zero.
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Last Time

• Conservation of Energy
• Escape velocity
• Power
• Potential energy diagrams

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Today

• Define linear momentum
• Relationship between K.E. and momentum
• More general form of 2nd law
• Impulse
• Internal and external forces
• Collisions

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New Concept Linear Momentum

• Linear momentum is simply the
• product of mass and velocity.
• Linear momentum is a vector.
• Sometimes we say just momentum.
• SI unit kg m/s

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Kinetic energy and linear momentum are intimately
related. Remember this result

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Do demo with bouncing ball and bean bag. One
recoils, the other doesnt.
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Change in Momentum

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Momentum and Newtons 2nd Law

• A more general form of Newtons 2nd law

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Lets see if the equations are consistent.
But when mass is not constant, our new general
form should be used.
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Impulse

• What is impulse and why is it useful?
• Forces sometimes act between objects over very
  short times.
• Examples
• Bouncing balls
• Bat hitting a ball
• Collisions

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The Average Force During a Collision
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Definition of Impulse

• Force can vary considerably over the time of
  interaction, so lets consider the average force,
  

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Impulse is just the change in momentum!
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Do egg in a sheet demo.
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Conservation of Linear Momentum

• What happens when 0?
• Then

If the net force acting on an object is zero, its
momentum is conserved.
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Do demos Rocket bicycle Reaction cars Fire
extinguisher rocket cart Water rocket 2-liter
bottle rocket
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Conservation of Linear Momentum
Law of conservation of linear momentum When the
net external force on a system of objects is
zero, the total momentum of the system remains
constant. Equivalently, The total momentum of an
isolated system remains constant.
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Conceptual Quiz

• An open cart rolls along a frictionless track
  while it is raining. As it rolls, what happens
  to the speed of the cart as the rain collects in
  it? (Assume that the rain falls vertically into
  the box.)

A) speeds up B) maintains constant speed C) slows
down D) stops immediately
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Conceptual Quiz

• An open cart rolls along a frictionless track
  while it is raining. As it rolls, what happens
  to the speed of the cart as the rain collects in
  it? (Assume that the rain falls vertically into
  the box.)

A) speeds up B) maintains constant speed C) slows
down D) stops immediately
Because the rain falls in vertically, it adds
no momentum to the box, thus the boxs momentum
is conserved. However, because the mass of the
box slowly increases with the added rain, its
velocity has to decrease.
Follow-up What happens to the cart when it
stops raining?
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Internal and External Forces

• If we have a system of particles, then there can
  be internal forces (for example, that hold the
  object together).
• Internal forces always occur in action-reaction
  pairs and the sum will be zero.

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System of Objects

• Internal forces have no effect on the net
  momentum of an object.
• If the net external force acting on a system is
  zero, then the net momentum is conserved.
• Momentum of every particle in system is not
  conserved, only the net.

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Conservation of Momentum
Momentum conservation works for a rocket as long
as we consider the rocket and its fuel to be one
system and account for the mass loss of the
rocket.
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Rocket Travel. A rocket of total mass 3180 kg
is traveling in outer space with a velocity of
115 m/s. To alter its course by 35.0, its
rockets can be fired briefly in a direction
perpendicular to its original motion. If the
rocket gases are expelled at a speed of 1750 m/s,
how much mass must be expelled?
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Conceptual Quiz
A system of particles is known to have a total
momentum of zero. Does it necessarily follow
that the total kinetic energy of the system is
also zero?
A) yes B) no
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Conceptual Quiz
A system of particles is known to have a total
momentum of zero. Does it necessarily follow
that the total kinetic energy of the system is
also zero?
A) yes B) no
Momentum is a vector, so the fact that ptot 0
does not mean that the particles are at rest!
They could be moving such that their momenta
cancel out when you add up all of the vectors.
In that case, because they are moving, the
particles would have non-zero KE.
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Conceptual Quiz

• Two boxes, one heavier than the other, are
  initially at rest on a horizontal frictionless
  surface. The same constant force F acts on each
  one for exactly 1 second. Which box has more
  momentum after the force acts ?

A) the heavier one B) the lighter one C) both
the same
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Conceptual Quiz

• Two boxes, one heavier than the other, are
  initially at rest on a horizontal frictionless
  surface. The same constant force F acts on each
  one for exactly 1 second. Which box has more
  momentum after the force acts ?

A) the heavier one B) the lighter one C) both
the same
,
We know
so impulse Dp Fav Dt.
In this case F and Dt are the same for both
boxes!
Both boxes will have the same final momentum.
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Conceptual Quiz
In the previous question, which box has the
larger velocity after the force acts?
A) the heavier one B) the lighter one C)
both the same
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Conceptual Quiz
In the previous question, which box has the
larger velocity after the force acts?
A) the heavier one B) the lighter one C)
both the same
The force is related to the acceleration by
Newtons Second Law (F ma). The lighter box
therefore has the greater acceleration and will
reach a higher speed after the 1-second time
interval.
Follow-up Which box has gone a larger distance
after the force acts?
Follow-up Which box has gained more KE after
the force acts?
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Conceptual Quiz

• A bowling ball and a Ping-Pong ball are rolling
  toward you with the same momentum. If you exert
  the same force to stop each one, which takes a
  longer time to bring to rest?

A) the bowling ball B) same time for both C)
the Ping-Pong ball D) impossible to say
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Conceptual Quiz

• A bowling ball and a Ping-Pong ball are rolling
  toward you with the same momentum. If you exert
  the same force to stop each one, which takes a
  longer time to bring to rest?

A) the bowling ball B) same time for both C)
the Ping-Pong ball D) impossible to say
We know
so Dp Fav Dt
Here, F and Dp are the same for both balls!
It will take the same amount of time to stop them.
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Conceptual Quiz

• A bowling ball and a Ping-Pong ball are rolling
  toward you with the same momentum. If you exert
  the same force to stop each one, for which is the
  stopping distance greater?

A) the bowling ball B) same distance for
both C) the Ping-Pong ball D) impossible to say
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Conceptual Quiz

• A bowling ball and a Ping-Pong ball are rolling
  toward you with the same momentum. If you exert
  the same force to stop each one, for which is the
  stopping distance greater?

A) the bowling ball B) same distance for
both C) the Ping-Pong ball D) impossible to say
Use the work-energy theorem W DKE. The
ball with less mass has the greater speed (why?),
and thus the greater KE (why again?). In order
to remove that KE, work must be done, where W
Fd. Because the force is the same in both cases,
the distance needed to stop the less massive ball
must be bigger.
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Conceptual Quiz
A) 2 m/s B) 6 m/s C) 9 m/s D) 12 m/s E) 18
m/s

• Amy (150 lbs) and Gwen (50 lbs) are standing on
  slippery ice and push off each other. If Amy
  slides at 6 m/s, what speed does Gwen have?

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Conceptual Quiz
A) 2 m/s B) 6 m/s C) 9 m/s D) 12 m/s E) 18
m/s

• Amy (150 lbs) and Gwen (50 lbs) are standing on
  slippery ice and push off each other. If Amy
  slides at 6 m/s, what speed does Gwen have?

The initial momentum is zero, so the momenta
of Amy and Gwen must be equal and opposite.
Because p mv, then if Amy has three times more
mass, we see that Gwen must have three times more
speed.