Ch. 8 Momentum and its conservation

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Ch. 8 Momentum and its conservation

• AP Physics C

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Citations

• Wahl, Michael. SparkNote on Conservation of
  Momentum. 25 Oct. 2008 lthttp//www.sparknotes.com/
  physics/linearmomentum/conservationofmomentumgt.
• Wahl, Michael. SparkNote on Collisions. 25 Oct.
  2008 lthttp//www.sparknotes.com/physics/linearmome
  ntum/collisionsgt.

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Terms

• Center of Mass  -  The point at which a given net
  force acting on the system will produce the same
  acceleration as if all the mass were concentrated
  at that point.
• Impulse  -  A force applied over a period of
  time.
• Momentum  -  The product of an object's mass and
  velocity.
• Conservation of Momentum  -  The principle
  stating that for any system with no external
  forces acting on it, momentum will be conserved.

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Center of mass

• We have been studying the mechanics of single
  particles.
• We will now expand our study to systems of
  several particles.
• The concept of the center of mass allows us to
  describe the movement of a system of particles by
  the movement of a single point.

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Center of mass equations

• X-coordinate of the center of mass
• Y-coordinate of the center of mass

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Sample problem 1

• Calculate the center of mass of the following
  system A mass of 5 kg lies at x 1, a mass of 3
  kg lies at x 4 and a mass of 2 kg lies at x
  0.

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Sample problem 2

• Calculate the center of mass of the following
  system A mass of 10 kg lies at the point (1,0),
  a mass of 2 kg lies at the point (2,1) and a mass
  of 5 kg lies at the point (0,1).

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Center of mass equations

• Velocity of the center of mass
• Acceleration of the center of mass
• External Net Force

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Sample problem 3

• Consider the system from problem 2, but now with
  forces acting upon the system. On the 10 kg mass,
  there is a force of 10 N in the positive x
  direction. On the 2 kg mass, there is a force of
  5 N inclined 45o above horizontal. Finally, on
  the 5 kg mass, there is a force of 2 N in the
  negative y direction. Find the resultant
  acceleration of the system.

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Sample problem 4

• Two masses, m1 and m2, m1 being larger, are
  connected by a spring. They are placed on a
  frictionless surface and separated so as to
  stretch the spring. They are then released from
  rest. In what direction does the system travel?

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Sample problem 5

• A 50 kg man stands at the edge of a raft of mass
  10 kg that is 10 meters long. The edge of the
  raft is against the shore of the lake. The man
  walks toward the shore, the entire length of the
  raft. How far from the shore does the raft move?

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Impulse

• We shall define this concept, force applied over
  a time period, as impulse. Impulse can be defined
  mathematically, and is denoted by J
• J F?t

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More on impulse

• It is a vector quantity.
• Can we predict the motion of an object?

• J F?t (ma)?t
• J m ?t
• J m?v ?(mv) mvf - mvo

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Sample problem 6

• What is the impulse of a force of 10 N acting on
  a ball for 2 seconds?
• The ball has a mass of 2 kg and is initially at
  rest. What is the velocity of the ball after the
  force has acted on it?

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Momentum

• From our equation relating impulse and velocity,
  it is logical to define the momentum of a single
  particle, denoted by the vector p, as such
• p mv    
• Again, momentum is a vector quantity, pointing in
  the direction of the velocity of the object. From
  this definition we can generate two very
  important equations, the first relating force and
  acceleration, the second relating impulse and
  momentum.

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Sample problem 7

• A particle has linear momentum of 10 kg-m/s, and
  a kinetic energy of 25 J. What is the mass of the
  particle?

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Relation between force and acceleration

• If we take a time derivative of our momentum
  expression we get the following equation
• Newtons Second Law of Motion

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Impulse-momentum theorem
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Sample problem 8

• A 2 kg bouncy ball is dropped from a height of 10
  meters, hits the floor and returns to its
  original height. What was the change in momentum
  of the ball upon impact with the floor? What was
  the impulse provided by the floor?

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Sample problem 9

• A ball of 2 kg is thrown straight up into the air
  with an initial velocity of 10 m/s. Using the
  impulse-momentum theorem, calculate the time of
  flight of the ball.

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Momentum and kinetic energy

• Recall
• Impulse is a change in momentum.
• Work done is a change in kinetic energy.

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Total momentum of a system

• Suppose we have a system of N particles, with
  masses m1, m2,, mn. Assuming no mass enters or
  leaves the system, we define the total momentum
  of the system as the vector sum of the individual
  momentum of the particles
• P p1 p2 ... pn   m1v1 m2v2 ...
  mnvn

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Total momentum of a system

• Recall
• vcm (m1v1 m2v2 ... mnvn)

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Conservation of momentum

• If the net external force is zero, then the total
  momentum of the system is constant.

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Conservation of momentum

• That is,
• Remember that momentum is a vector quantity.

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Sample problem 10

• A 60 kg man standing on a stationary 40 kg boat
  throws a .2 kg baseball with a velocity of 50
  m/s. With what speed does the boat move after the
  man throws the ball?

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Sample problem 11

• A .05 kg bullet is fired at a velocity of 500
  m/s, and embeds itself in a block of mass 4 kg,
  initially at rest and on a frictionless surface.
  What is the final velocity of the block?

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Sample problem 12

• An object at rest explodes into three pieces.
  Two, each of the same mass, fly off in different
  directions with velocity 50 m/s and 100 m/s,
  respectively. A third piece goes off in the
  negative y-direction is also formed in the
  explosion, and has twice the mass of the first
  two pieces. Determine the direction of the second
  particle and the speed of the third particle.
  Let ?1 65o.

?2
T1 65o
v3
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Sample problem 13

• A spaceship moving at 1000 m/s fires a missile of
  mass 1000 kg at a speed of 10000 m/s. What is the
  mass of the spaceship it slows down to a velocity
  of 910 m/s?

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Collisions

• Collision  -  The brief direct contact between
  two bodies that results in a net impulse on each
  body.
• Elastic Collision  -  Any collision in which
  kinetic energy is conserved.
• Inelastic Collision  -  Any collision in which
  kinetic energy is not conserved.
• Completely Inelastic Collision  -  Any collision
  in which the two bodies stick together.

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Elastic collisions

• Why are these collisions special? We know with
  all collisions that momentum is conserved. If two
  particles collide we can use the following
  equation
• m1v1o m2v2o m1v1f m2v2f
• However, we also know that, because the collision
  is elastic, kinetic energy is conserved. For the
  same situation we can use the following equation
  
• m1v1o2 m2v2o2 m1v1f2 m2v2f2

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Sample problem 14

• Two balls, each with mass 2 kg, and velocities of
  2 m/s and 3 m/s collide head on. Their final
  velocities are 2 m/s and 1 m/s, respectively. Is
  this collision elastic or inelastic?

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Sample problem 15

• Two balls of mass m1 and m2, with velocities v1
  and v2 collide head on. Is there any way for both
  balls to have zero velocity after the collision?
  If so, find the conditions under which this can
  occur.

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Sample problem 16

• Two balls with equal masses, m, and equal speed,
  v, engage in a head on elastic collision. What is
  the final velocity of each ball, in terms of m
  and v?

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Sample problem 17

• One pool ball traveling with a velocity of 5 m/s
  hits another ball of the same mass, which is
  stationary. The collision is head on and elastic.
  Find the final velocities of both balls.

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Inelastic collisions

• So what if kinetic energy is not conserved? Our
  knowledge of such situations is more limited,
  since we no longer know what the kinetic energy
  is after the collision. However, even though
  kinetic energy is not conserved, momentum will
  always be conserved.

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Completely inelastic collision

• Consider the case in which two particles collide,
  and actually physically stick together. In this
  case, called a completely inelastic collision we
  only need to solve for one final velocity, and
  the conservation of momentum equation is enough
  to predict the outcome of the collision. The two
  particles in a completely inelastic collision
  must move at the same final velocity, so our
  linear momentum equation becomes
• m1v1o m2v2o m1vf m2vf    
• Thus
• m1v1o m2v2o Mvf    

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Sample problem 18

• A car of 500 kg, traveling at 30 m/s rear ends
  another car of 600 kg, traveling at 20 m/s. in
  the same direction The collision is great enough
  that the two cars stick together after they
  collide. How fast will both cars be going after
  the collision?

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2-d collisions

• Two balls of equal masses move toward each other
  on the x-axis. When they collide, each ball
  ricochets 90 degrees, such that both balls are
  moving away from each other on the y-axis. What
  can be said about the final velocity of each
  ball?

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Sample problem 19

• Two pool balls traveling in opposite directions
  collide. One ball travels off at an angle ? to
  its original velocity, as shown below. Is there
  any possible way for the second ball to be
  completely stopped by this collision? If so state
  the conditions under which this could occur.

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Sample problem 20

• Two objects are traveling perpendicular to each
  other, one moving at 2 m/s with a mass of 5 kg,
  and one moving at 3 m/s with a mass of 10 kg, as
  shown below. They collide and stick together.
  What is the magnitude and direction of the
  velocity of both objects?

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Sample problem 21

• A common pool shot involves hitting a ball into a
  pocket from an angle. Shown below, the cue ball
  hits a stationary ball at an angle of 45o, such
  that it goes into the corner pocket with a speed
  of 2 m/s. Both balls have a mass of .5 kg, and
  the cue ball is traveling at 4 m/s before the
  collision. Calculate the angle with which the cue
  is deflected by the collision.